{"id":4522,"date":"2024-10-04T12:40:54","date_gmt":"2024-10-04T12:40:54","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=4522"},"modified":"2024-11-21T18:50:12","modified_gmt":"2024-11-21T18:50:12","slug":"rational-and-radical-functions-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/rational-and-radical-functions-get-stronger\/","title":{"raw":"Rational and Radical Functions: Get Stronger","rendered":"Rational and Radical Functions: Get Stronger"},"content":{"raw":"

Rational Functions<\/span><\/h2>\r\n

For the following exercises, find the domain of the rational functions.<\/p>\r\n\r\n

    \r\n \t
  1. [latex]f(x)=\\dfrac{x+1}{x^2-1}[\/latex]<\/li>\r\n \t
  2. [latex]f(x)=\\dfrac{x^2+4x-3}{x^4-5x^2+4}[\/latex]<\/li>\r\n<\/ol>\r\n

    For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.<\/p>\r\n\r\n

      \r\n \t
    1. [latex]f(x)=\\dfrac{2}{5x+2}[\/latex]<\/li>\r\n \t
    2. [latex]f(x)=\\dfrac{x}{x^2+5x-36}[\/latex]<\/li>\r\n \t
    3. [latex]f(x)=\\dfrac{3x-4}{x^3-16x}[\/latex]<\/li>\r\n \t
    4. [latex]f(x)=\\dfrac{x+5}{x^2-25}[\/latex]<\/li>\r\n \t
    5. [latex]f(x)=\\dfrac{4-2x}{3x-1}[\/latex]<\/li>\r\n<\/ol>\r\n

      For the following exercises, find the x- and y-intercepts for the functions.<\/p>\r\n\r\n

        \r\n \t
      1. [latex]f(x)=\\dfrac{x}{x^2-x}[\/latex]<\/li>\r\n \t
      2. [latex]f(x)=\\dfrac{x^2+x+6}{x^2-10x+24}[\/latex]<\/li>\r\n<\/ol>\r\n

        For the following exercises, describe the local and end behavior of the functions.<\/p>\r\n\r\n

          \r\n \t
        1. [latex]f(x)=\\dfrac{x}{2x+1}[\/latex]<\/li>\r\n \t
        2. [latex]f(x)=\\dfrac{-2x}{x-6}[\/latex]<\/li>\r\n \t
        3. [latex]f(x)=\\dfrac{2x^2-32}{6x^2+13x-5}[\/latex]<\/li>\r\n<\/ol>\r\n

          For the following exercises, find the slant asymptote of the functions.<\/p>\r\n\r\n

            \r\n \t
          1. [latex]f(x)=\\dfrac{4x^2-10}{2x-4}[\/latex]<\/li>\r\n \t
          2. [latex]f(x)=\\dfrac{6x^3-5x}{3x^2+4}[\/latex]<\/li>\r\n<\/ol>\r\n

            For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph.<\/p>\r\n\r\n

              \r\n \t
            1. [latex]p(x)=\\dfrac{2x-3}{x+4}[\/latex]<\/li>\r\n \t
            2. [latex]s(x)=\\dfrac{4}{(x-2)^2}[\/latex]<\/li>\r\n \t
            3. [latex]f(x)=\\dfrac{3x^2-14x-5}{3x^2+8x-16}[\/latex]<\/li>\r\n \t
            4. [latex]a(x)=\\dfrac{x^2+2x-3}{x^2-1}[\/latex]<\/li>\r\n<\/ol>\r\n

              For the following exercises, write an equation for a rational function with the given characteristics.<\/p>\r\n\r\n

                \r\n \t
              1. Vertical asymptotes at [latex]x=5[\/latex] and [latex]x=-5[\/latex], [latex]x[\/latex]-intercepts at [latex](2,0)[\/latex] and [latex](-1,0)[\/latex], [latex]y[\/latex]-intercept at [latex](0,4)[\/latex]<\/li>\r\n \t
              2. Vertical asymptotes at [latex]x=-4[\/latex] and [latex]x=-5[\/latex], [latex]x[\/latex]-intercepts at [latex](4,0)[\/latex] and [latex](-6,0)[\/latex], Horizontal asymptote at [latex]y=7[\/latex]<\/li>\r\n \t
              3. Vertical asymptote at [latex]x=-1[\/latex], Double zero at [latex]x=2[\/latex], [latex]y[\/latex]-intercept at [latex](0,2)[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, use the graphs to write an equation for the function.\r\n
                  \r\n \t
                1. \"Graph<\/li>\r\n \t
                2. \"Graph<\/li>\r\n<\/ol>\r\n

                  For the following exercise, express a rational function that describes the situation.<\/p>\r\n\r\n

                    \r\n \t
                  1. In the refugee camp hospital, a large mixing tank currently contains [latex]300[\/latex] gallons of water, into which [latex]8[\/latex] pounds of sugar have been mixed. A tap will open, pouring [latex]20[\/latex] gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of [latex]2[\/latex] pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after [latex]t[\/latex] minutes.<\/li>\r\n<\/ol>\r\n

                    For the following exercise, use the given rational function to answer the question.<\/p>\r\n\r\n

                      \r\n \t
                    1. The concentration [latex]C[\/latex] of a drug in a patient's bloodstream [latex]t[\/latex] hours after injection is given by [latex]C(t)=\\dfrac{100t}{2t^2+75}[\/latex]. Use a calculator to approximate the time when the concentration is highest.<\/li>\r\n<\/ol>\r\n

                      For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question.<\/p>\r\n\r\n

                        \r\n \t
                      1. A rectangular box with a square base is to have a volume of [latex]20[\/latex] cubic feet. The material for the base costs [latex]30[\/latex] cents\/square foot. The material for the sides costs [latex]10[\/latex] cents\/square foot. The material for the top costs [latex]20[\/latex] cents\/square foot. Determine the dimensions that will yield minimum cost. Let [latex]x[\/latex] = length of the side of the base.<\/li>\r\n \t
                      2. A right circular cylinder with no top has a volume of [latex]50[\/latex] cubic meters. Find the radius that will yield minimum surface area. Let [latex]x[\/latex] = radius.<\/li>\r\n<\/ol>\r\n

                        Inverses and Radical Functions<\/span><\/h2>\r\n

                        For the following exercises, find the inverse of the function on the given domain.<\/p>\r\n\r\n

                          \r\n \t
                        1. [latex]f(x)=(x-4)^2, [4,\\infty)[\/latex]<\/li>\r\n \t
                        2. [latex]f(x)=(x+1)^2-3, [-1,\\infty)[\/latex]<\/li>\r\n \t
                        3. [latex]f(x)=12-x^2, [0,\\infty)[\/latex]<\/li>\r\n \t
                        4. [latex]f(x)=2x^2+4, [0,\\infty)[\/latex]<\/li>\r\n<\/ol>\r\n

                          For the following exercises, find the inverse of the functions.<\/p>\r\n\r\n

                            \r\n \t
                          1. [latex]f(x)=3x^3+1[\/latex]<\/li>\r\n \t
                          2. [latex]f(x)=4-2x^3[\/latex]<\/li>\r\n<\/ol>\r\n

                            For the following exercises, find the inverse of the functions.<\/p>\r\n\r\n

                              \r\n \t
                            1. [latex]f(x)=\\sqrt{3-4x}[\/latex]<\/li>\r\n \t
                            2. [latex]f(x)=\\sqrt{6x-8}+5[\/latex]<\/li>\r\n \t
                            3. [latex]f(x)=3-\\sqrt[3]{x}[\/latex]<\/li>\r\n \t
                            4. [latex]f(x)=\\dfrac{3}{x-4}[\/latex]<\/li>\r\n \t
                            5. [latex]f(x)=\\dfrac{x-2}{x+7}[\/latex]<\/li>\r\n \t
                            6. [latex]f(x)=\\dfrac{5x+1}{2-5x}[\/latex]<\/li>\r\n \t
                            7. [latex]f(x)=x^2+4x+1, [-2,\\infty)[\/latex]<\/li>\r\n<\/ol>\r\n

                              For the following exercises, find the inverse of the function and graph both the function and its inverse.<\/p>\r\n\r\n

                                \r\n \t
                              1. [latex]f(x)=x^2+2, x \\geq 0[\/latex]<\/li>\r\n \t
                              2. [latex]f(x)=(x+3)^2, x \\geq -3[\/latex]<\/li>\r\n \t
                              3. [latex]f(x)=x^3+3[\/latex]<\/li>\r\n<\/ol>\r\n

                                For the following exercises, use a graph to help determine the domain of the functions.<\/p>\r\n\r\n

                                  \r\n \t
                                1. [latex]f(x)=\\sqrt{\\dfrac{(x+1)(x-1)}{x}}[\/latex]<\/li>\r\n \t
                                2. [latex]f(x)=\\sqrt{\\dfrac{x(x+3)}{x-4}}[\/latex]<\/li>\r\n \t
                                3. [latex]f(x)=\\sqrt{\\dfrac{9-x^2}{x+4}}[\/latex]<\/li>\r\n<\/ol>\r\n

                                  For the following exercises, determine the function described and then use it to answer the question.<\/p>\r\n\r\n

                                    \r\n \t
                                  1. An object dropped from a height of [latex]600[\/latex] feet has a height, [latex]h(t)[\/latex], in feet after [latex]t[\/latex] seconds have elapsed, such that [latex]h(t)=600-16t^2[\/latex]. Express [latex]t[\/latex] as a function of height [latex]h[\/latex], and find the time to reach a height of [latex]400[\/latex] feet.<\/li>\r\n \t
                                  2. The surface area, [latex]A[\/latex], of a sphere in terms of its radius, [latex]r[\/latex], is given by [latex]A(r)=4\\pi r^2[\/latex]. Express [latex]r[\/latex] as a function of [latex]A[\/latex], and find the radius of a sphere with a surface area of [latex]1000[\/latex] square inches.<\/li>\r\n \t
                                  3. The period [latex]T[\/latex], in seconds, of a simple pendulum as a function of its length [latex]l[\/latex], in feet, is given by [latex]T(l)=2\\pi\\sqrt{\\dfrac{l}{32.2}}[\/latex]. Express [latex]l[\/latex] as a function of [latex]T[\/latex] and determine the length of a pendulum with period of [latex]2[\/latex] seconds.<\/li>\r\n \t
                                  4. The surface area, [latex]A[\/latex], of a cylinder in terms of its radius, [latex]r[\/latex], and height, [latex]h[\/latex], is given by [latex]A=2\\pi r^2+2\\pi rh[\/latex]. If the height of the cylinder is [latex]4[\/latex] feet, express the radius as a function of [latex]A[\/latex] and find the radius if the surface area is [latex]200[\/latex] square feet.<\/li>\r\n \t
                                  5. Consider a cone with height of [latex]30[\/latex] feet. Express the radius, [latex]r[\/latex], in terms of the volume, [latex]V[\/latex], and find the radius of a cone with volume of [latex]1000[\/latex] cubic feet.<\/li>\r\n<\/ol>\r\n

                                    Variations<\/span><\/h2>\r\n

                                    For the following exercises, write an equation describing the relationship of the given variables.<\/p>\r\n\r\n

                                      \r\n \t
                                    1. [latex]y[\/latex] varies directly as the square of [latex]x[\/latex] and when [latex]x=4[\/latex], [latex]y=80[\/latex].<\/li>\r\n \t
                                    2. [latex]y[\/latex] varies directly as the cube of [latex]x[\/latex] and when [latex]x=36[\/latex], [latex]y=24[\/latex].<\/li>\r\n \t
                                    3. [latex]y[\/latex] varies directly as the fourth power of [latex]x[\/latex] and when [latex]x=1[\/latex], [latex]y=6[\/latex].<\/li>\r\n \t
                                    4. [latex]y[\/latex] varies inversely as the square of [latex]x[\/latex] and when [latex]x=3[\/latex], [latex]y=2[\/latex].<\/li>\r\n \t
                                    5. [latex]y[\/latex] varies inversely as the fourth power of [latex]x[\/latex] and when [latex]x=3[\/latex], [latex]y=1[\/latex].<\/li>\r\n \t
                                    6. [latex]y[\/latex] varies inversely as the cube root of [latex]x[\/latex] and when [latex]x=64[\/latex], [latex]y=5[\/latex].<\/li>\r\n \t
                                    7. [latex]y[\/latex] varies jointly as [latex]x[\/latex], [latex]z[\/latex], and [latex]w[\/latex] and when [latex]x=1[\/latex], [latex]z=2[\/latex], [latex]w=5[\/latex], then [latex]y=100[\/latex].<\/li>\r\n \t
                                    8. [latex]y[\/latex] varies jointly as [latex]x[\/latex] and the square root of [latex]z[\/latex] and when [latex]x=2[\/latex] and [latex]z=25[\/latex], then [latex]y=100[\/latex].<\/li>\r\n \t
                                    9. [latex]y[\/latex] varies jointly as [latex]x[\/latex] and [latex]z[\/latex] and inversely as [latex]w[\/latex]. When [latex]x=3[\/latex], [latex]z=5[\/latex], and [latex]w=6[\/latex], then [latex]y=10[\/latex].<\/li>\r\n \t
                                    10. [latex]y[\/latex] varies jointly as [latex]x[\/latex] and [latex]z[\/latex] and inversely as the square root of [latex]w[\/latex] and the square of [latex]t[\/latex]. When [latex]x=3[\/latex], [latex]z=1[\/latex], [latex]w=25[\/latex], and [latex]t=2[\/latex], then [latex]y=6[\/latex].<\/li>\r\n<\/ol>\r\n

                                      For the following exercises, use the given information to find the unknown value.<\/p>\r\n\r\n

                                        \r\n \t
                                      1. [latex]y[\/latex] varies directly as the square of [latex]x[\/latex]. When [latex]x=2[\/latex], then [latex]y=16[\/latex]. Find [latex]y[\/latex] when [latex]x=8[\/latex].<\/li>\r\n \t
                                      2. [latex]y[\/latex] varies directly as the square root of [latex]x[\/latex]. When [latex]x=16[\/latex], then [latex]y=4[\/latex]. Find [latex]y[\/latex] when [latex]x=36[\/latex].<\/li>\r\n \t
                                      3. [latex]y[\/latex] varies inversely with [latex]x[\/latex]. When [latex]x=3[\/latex], then [latex]y=2[\/latex]. Find [latex]y[\/latex] when [latex]x=1[\/latex].<\/li>\r\n \t
                                      4. [latex]y[\/latex] varies inversely with the cube of [latex]x[\/latex]. When [latex]x=3[\/latex], then [latex]y=1[\/latex]. Find [latex]y[\/latex] when [latex]x=1[\/latex].<\/li>\r\n \t
                                      5. [latex]y[\/latex] varies inversely with the cube root of [latex]x[\/latex]. When [latex]x=27[\/latex], then [latex]y=5[\/latex]. Find [latex]y[\/latex] when [latex]x=125[\/latex].<\/li>\r\n \t
                                      6. [latex]y[\/latex] varies jointly as [latex]x[\/latex], [latex]z[\/latex], and [latex]w[\/latex]. When [latex]x=2[\/latex], [latex]z=1[\/latex], and [latex]w=12[\/latex], then [latex]y=72[\/latex]. Find [latex]y[\/latex] when [latex]x=1[\/latex], [latex]z=2[\/latex], and [latex]w=3[\/latex].<\/li>\r\n \t
                                      7. [latex]y[\/latex] varies jointly as [latex]x[\/latex] and [latex]z[\/latex] and inversely as [latex]w[\/latex]. When [latex]x=5[\/latex], [latex]z=2[\/latex], and [latex]w=20[\/latex], then [latex]y=4[\/latex]. Find [latex]y[\/latex] when [latex]x=3[\/latex] and [latex]z=8[\/latex], and [latex]w=48[\/latex].<\/li>\r\n \t
                                      8. [latex]y[\/latex] varies jointly as the square of [latex]x[\/latex] and the cube of [latex]z[\/latex] and inversely as the square root of [latex]w[\/latex]. When [latex]x=2[\/latex], [latex]z=2[\/latex], and [latex]w=64[\/latex], then [latex]y=12[\/latex]. Find [latex]y[\/latex] when [latex]x=1[\/latex], [latex]z=3[\/latex], and [latex]w=4[\/latex].<\/li>\r\n<\/ol>\r\n

                                        For the following exercises, use the given information to answer the questions.<\/p>\r\n\r\n

                                          \r\n \t
                                        1. The distance [latex]s[\/latex] that an object falls varies directly with the square of the time, [latex]t[\/latex], of the fall. If an object falls [latex]16[\/latex] feet in [latex]1[\/latex] second, how long for it to fall [latex]144[\/latex] feet?<\/li>\r\n \t
                                        2. The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is [latex]24[\/latex] inches long and vibrates [latex]128[\/latex] times per second, what is the length of a string that vibrates [latex]64[\/latex] times per second?<\/li>\r\n \t
                                        3. The weight of an object above the surface of Earth varies inversely with the square of the distance from the center of Earth. If a body weighs [latex]50[\/latex] pounds when it is [latex]3960[\/latex] miles from Earth's center, what would it weigh if it were [latex]3970[\/latex] miles from Earth's center?<\/li>\r\n \t
                                        4. The current in a circuit varies inversely with its resistance measured in ohms. When the current in a circuit is [latex]40[\/latex] amperes, the resistance is [latex]10[\/latex] ohms. Find the current if the resistance is [latex]12[\/latex] ohms.<\/li>\r\n \t
                                        5. The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm) and the cube of the diameter. If the shaft of a certain material [latex]3[\/latex] inches in diameter can transmit [latex]45[\/latex] hp at [latex]100[\/latex] rpm, what must the diameter be in order to transmit [latex]60[\/latex] hp at [latex]150[\/latex] rpm?<\/li>\r\n<\/ol>","rendered":"

                                          Rational Functions<\/span><\/h2>\n

                                          For the following exercises, find the domain of the rational functions.<\/p>\n

                                            \n
                                          1. [latex]f(x)=\\dfrac{x+1}{x^2-1}[\/latex]<\/li>\n
                                          2. [latex]f(x)=\\dfrac{x^2+4x-3}{x^4-5x^2+4}[\/latex]<\/li>\n<\/ol>\n

                                            For the following exercises, find the domain, vertical asymptotes, and horizontal asymptotes of the functions.<\/p>\n

                                              \n
                                            1. [latex]f(x)=\\dfrac{2}{5x+2}[\/latex]<\/li>\n
                                            2. [latex]f(x)=\\dfrac{x}{x^2+5x-36}[\/latex]<\/li>\n
                                            3. [latex]f(x)=\\dfrac{3x-4}{x^3-16x}[\/latex]<\/li>\n
                                            4. [latex]f(x)=\\dfrac{x+5}{x^2-25}[\/latex]<\/li>\n
                                            5. [latex]f(x)=\\dfrac{4-2x}{3x-1}[\/latex]<\/li>\n<\/ol>\n

                                              For the following exercises, find the x- and y-intercepts for the functions.<\/p>\n

                                                \n
                                              1. [latex]f(x)=\\dfrac{x}{x^2-x}[\/latex]<\/li>\n
                                              2. [latex]f(x)=\\dfrac{x^2+x+6}{x^2-10x+24}[\/latex]<\/li>\n<\/ol>\n

                                                For the following exercises, describe the local and end behavior of the functions.<\/p>\n

                                                  \n
                                                1. [latex]f(x)=\\dfrac{x}{2x+1}[\/latex]<\/li>\n
                                                2. [latex]f(x)=\\dfrac{-2x}{x-6}[\/latex]<\/li>\n
                                                3. [latex]f(x)=\\dfrac{2x^2-32}{6x^2+13x-5}[\/latex]<\/li>\n<\/ol>\n

                                                  For the following exercises, find the slant asymptote of the functions.<\/p>\n

                                                    \n
                                                  1. [latex]f(x)=\\dfrac{4x^2-10}{2x-4}[\/latex]<\/li>\n
                                                  2. [latex]f(x)=\\dfrac{6x^3-5x}{3x^2+4}[\/latex]<\/li>\n<\/ol>\n

                                                    For the following exercises, find the horizontal intercepts, the vertical intercept, the vertical asymptotes, and the horizontal or slant asymptote of the functions. Use that information to sketch a graph.<\/p>\n

                                                      \n
                                                    1. [latex]p(x)=\\dfrac{2x-3}{x+4}[\/latex]<\/li>\n
                                                    2. [latex]s(x)=\\dfrac{4}{(x-2)^2}[\/latex]<\/li>\n
                                                    3. [latex]f(x)=\\dfrac{3x^2-14x-5}{3x^2+8x-16}[\/latex]<\/li>\n
                                                    4. [latex]a(x)=\\dfrac{x^2+2x-3}{x^2-1}[\/latex]<\/li>\n<\/ol>\n

                                                      For the following exercises, write an equation for a rational function with the given characteristics.<\/p>\n

                                                        \n
                                                      1. Vertical asymptotes at [latex]x=5[\/latex] and [latex]x=-5[\/latex], [latex]x[\/latex]-intercepts at [latex](2,0)[\/latex] and [latex](-1,0)[\/latex], [latex]y[\/latex]-intercept at [latex](0,4)[\/latex]<\/li>\n
                                                      2. Vertical asymptotes at [latex]x=-4[\/latex] and [latex]x=-5[\/latex], [latex]x[\/latex]-intercepts at [latex](4,0)[\/latex] and [latex](-6,0)[\/latex], Horizontal asymptote at [latex]y=7[\/latex]<\/li>\n
                                                      3. Vertical asymptote at [latex]x=-1[\/latex], Double zero at [latex]x=2[\/latex], [latex]y[\/latex]-intercept at [latex](0,2)[\/latex]<\/li>\n<\/ol>\n

                                                        For the following exercises, use the graphs to write an equation for the function.<\/p>\n

                                                          \n
                                                        1. \"Graph<\/li>\n
                                                        2. \"Graph<\/li>\n<\/ol>\n

                                                          For the following exercise, express a rational function that describes the situation.<\/p>\n

                                                            \n
                                                          1. In the refugee camp hospital, a large mixing tank currently contains [latex]300[\/latex] gallons of water, into which [latex]8[\/latex] pounds of sugar have been mixed. A tap will open, pouring [latex]20[\/latex] gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of [latex]2[\/latex] pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after [latex]t[\/latex] minutes.<\/li>\n<\/ol>\n

                                                            For the following exercise, use the given rational function to answer the question.<\/p>\n

                                                              \n
                                                            1. The concentration [latex]C[\/latex] of a drug in a patient’s bloodstream [latex]t[\/latex] hours after injection is given by [latex]C(t)=\\dfrac{100t}{2t^2+75}[\/latex]. Use a calculator to approximate the time when the concentration is highest.<\/li>\n<\/ol>\n

                                                              For the following exercises, construct a rational function that will help solve the problem. Then, use a calculator to answer the question.<\/p>\n

                                                                \n
                                                              1. A rectangular box with a square base is to have a volume of [latex]20[\/latex] cubic feet. The material for the base costs [latex]30[\/latex] cents\/square foot. The material for the sides costs [latex]10[\/latex] cents\/square foot. The material for the top costs [latex]20[\/latex] cents\/square foot. Determine the dimensions that will yield minimum cost. Let [latex]x[\/latex] = length of the side of the base.<\/li>\n
                                                              2. A right circular cylinder with no top has a volume of [latex]50[\/latex] cubic meters. Find the radius that will yield minimum surface area. Let [latex]x[\/latex] = radius.<\/li>\n<\/ol>\n

                                                                Inverses and Radical Functions<\/span><\/h2>\n

                                                                For the following exercises, find the inverse of the function on the given domain.<\/p>\n

                                                                  \n
                                                                1. [latex]f(x)=(x-4)^2, [4,\\infty)[\/latex]<\/li>\n
                                                                2. [latex]f(x)=(x+1)^2-3, [-1,\\infty)[\/latex]<\/li>\n
                                                                3. [latex]f(x)=12-x^2, [0,\\infty)[\/latex]<\/li>\n
                                                                4. [latex]f(x)=2x^2+4, [0,\\infty)[\/latex]<\/li>\n<\/ol>\n

                                                                  For the following exercises, find the inverse of the functions.<\/p>\n

                                                                    \n
                                                                  1. [latex]f(x)=3x^3+1[\/latex]<\/li>\n
                                                                  2. [latex]f(x)=4-2x^3[\/latex]<\/li>\n<\/ol>\n

                                                                    For the following exercises, find the inverse of the functions.<\/p>\n

                                                                      \n
                                                                    1. [latex]f(x)=\\sqrt{3-4x}[\/latex]<\/li>\n
                                                                    2. [latex]f(x)=\\sqrt{6x-8}+5[\/latex]<\/li>\n
                                                                    3. [latex]f(x)=3-\\sqrt[3]{x}[\/latex]<\/li>\n
                                                                    4. [latex]f(x)=\\dfrac{3}{x-4}[\/latex]<\/li>\n
                                                                    5. [latex]f(x)=\\dfrac{x-2}{x+7}[\/latex]<\/li>\n
                                                                    6. [latex]f(x)=\\dfrac{5x+1}{2-5x}[\/latex]<\/li>\n
                                                                    7. [latex]f(x)=x^2+4x+1, [-2,\\infty)[\/latex]<\/li>\n<\/ol>\n

                                                                      For the following exercises, find the inverse of the function and graph both the function and its inverse.<\/p>\n

                                                                        \n
                                                                      1. [latex]f(x)=x^2+2, x \\geq 0[\/latex]<\/li>\n
                                                                      2. [latex]f(x)=(x+3)^2, x \\geq -3[\/latex]<\/li>\n
                                                                      3. [latex]f(x)=x^3+3[\/latex]<\/li>\n<\/ol>\n

                                                                        For the following exercises, use a graph to help determine the domain of the functions.<\/p>\n

                                                                          \n
                                                                        1. [latex]f(x)=\\sqrt{\\dfrac{(x+1)(x-1)}{x}}[\/latex]<\/li>\n
                                                                        2. [latex]f(x)=\\sqrt{\\dfrac{x(x+3)}{x-4}}[\/latex]<\/li>\n
                                                                        3. [latex]f(x)=\\sqrt{\\dfrac{9-x^2}{x+4}}[\/latex]<\/li>\n<\/ol>\n

                                                                          For the following exercises, determine the function described and then use it to answer the question.<\/p>\n

                                                                            \n
                                                                          1. An object dropped from a height of [latex]600[\/latex] feet has a height, [latex]h(t)[\/latex], in feet after [latex]t[\/latex] seconds have elapsed, such that [latex]h(t)=600-16t^2[\/latex]. Express [latex]t[\/latex] as a function of height [latex]h[\/latex], and find the time to reach a height of [latex]400[\/latex] feet.<\/li>\n
                                                                          2. The surface area, [latex]A[\/latex], of a sphere in terms of its radius, [latex]r[\/latex], is given by [latex]A(r)=4\\pi r^2[\/latex]. Express [latex]r[\/latex] as a function of [latex]A[\/latex], and find the radius of a sphere with a surface area of [latex]1000[\/latex] square inches.<\/li>\n
                                                                          3. The period [latex]T[\/latex], in seconds, of a simple pendulum as a function of its length [latex]l[\/latex], in feet, is given by [latex]T(l)=2\\pi\\sqrt{\\dfrac{l}{32.2}}[\/latex]. Express [latex]l[\/latex] as a function of [latex]T[\/latex] and determine the length of a pendulum with period of [latex]2[\/latex] seconds.<\/li>\n
                                                                          4. The surface area, [latex]A[\/latex], of a cylinder in terms of its radius, [latex]r[\/latex], and height, [latex]h[\/latex], is given by [latex]A=2\\pi r^2+2\\pi rh[\/latex]. If the height of the cylinder is [latex]4[\/latex] feet, express the radius as a function of [latex]A[\/latex] and find the radius if the surface area is [latex]200[\/latex] square feet.<\/li>\n
                                                                          5. Consider a cone with height of [latex]30[\/latex] feet. Express the radius, [latex]r[\/latex], in terms of the volume, [latex]V[\/latex], and find the radius of a cone with volume of [latex]1000[\/latex] cubic feet.<\/li>\n<\/ol>\n

                                                                            Variations<\/span><\/h2>\n

                                                                            For the following exercises, write an equation describing the relationship of the given variables.<\/p>\n

                                                                              \n
                                                                            1. [latex]y[\/latex] varies directly as the square of [latex]x[\/latex] and when [latex]x=4[\/latex], [latex]y=80[\/latex].<\/li>\n
                                                                            2. [latex]y[\/latex] varies directly as the cube of [latex]x[\/latex] and when [latex]x=36[\/latex], [latex]y=24[\/latex].<\/li>\n
                                                                            3. [latex]y[\/latex] varies directly as the fourth power of [latex]x[\/latex] and when [latex]x=1[\/latex], [latex]y=6[\/latex].<\/li>\n
                                                                            4. [latex]y[\/latex] varies inversely as the square of [latex]x[\/latex] and when [latex]x=3[\/latex], [latex]y=2[\/latex].<\/li>\n
                                                                            5. [latex]y[\/latex] varies inversely as the fourth power of [latex]x[\/latex] and when [latex]x=3[\/latex], [latex]y=1[\/latex].<\/li>\n
                                                                            6. [latex]y[\/latex] varies inversely as the cube root of [latex]x[\/latex] and when [latex]x=64[\/latex], [latex]y=5[\/latex].<\/li>\n
                                                                            7. [latex]y[\/latex] varies jointly as [latex]x[\/latex], [latex]z[\/latex], and [latex]w[\/latex] and when [latex]x=1[\/latex], [latex]z=2[\/latex], [latex]w=5[\/latex], then [latex]y=100[\/latex].<\/li>\n
                                                                            8. [latex]y[\/latex] varies jointly as [latex]x[\/latex] and the square root of [latex]z[\/latex] and when [latex]x=2[\/latex] and [latex]z=25[\/latex], then [latex]y=100[\/latex].<\/li>\n
                                                                            9. [latex]y[\/latex] varies jointly as [latex]x[\/latex] and [latex]z[\/latex] and inversely as [latex]w[\/latex]. When [latex]x=3[\/latex], [latex]z=5[\/latex], and [latex]w=6[\/latex], then [latex]y=10[\/latex].<\/li>\n
                                                                            10. [latex]y[\/latex] varies jointly as [latex]x[\/latex] and [latex]z[\/latex] and inversely as the square root of [latex]w[\/latex] and the square of [latex]t[\/latex]. When [latex]x=3[\/latex], [latex]z=1[\/latex], [latex]w=25[\/latex], and [latex]t=2[\/latex], then [latex]y=6[\/latex].<\/li>\n<\/ol>\n

                                                                              For the following exercises, use the given information to find the unknown value.<\/p>\n

                                                                                \n
                                                                              1. [latex]y[\/latex] varies directly as the square of [latex]x[\/latex]. When [latex]x=2[\/latex], then [latex]y=16[\/latex]. Find [latex]y[\/latex] when [latex]x=8[\/latex].<\/li>\n
                                                                              2. [latex]y[\/latex] varies directly as the square root of [latex]x[\/latex]. When [latex]x=16[\/latex], then [latex]y=4[\/latex]. Find [latex]y[\/latex] when [latex]x=36[\/latex].<\/li>\n
                                                                              3. [latex]y[\/latex] varies inversely with [latex]x[\/latex]. When [latex]x=3[\/latex], then [latex]y=2[\/latex]. Find [latex]y[\/latex] when [latex]x=1[\/latex].<\/li>\n
                                                                              4. [latex]y[\/latex] varies inversely with the cube of [latex]x[\/latex]. When [latex]x=3[\/latex], then [latex]y=1[\/latex]. Find [latex]y[\/latex] when [latex]x=1[\/latex].<\/li>\n
                                                                              5. [latex]y[\/latex] varies inversely with the cube root of [latex]x[\/latex]. When [latex]x=27[\/latex], then [latex]y=5[\/latex]. Find [latex]y[\/latex] when [latex]x=125[\/latex].<\/li>\n
                                                                              6. [latex]y[\/latex] varies jointly as [latex]x[\/latex], [latex]z[\/latex], and [latex]w[\/latex]. When [latex]x=2[\/latex], [latex]z=1[\/latex], and [latex]w=12[\/latex], then [latex]y=72[\/latex]. Find [latex]y[\/latex] when [latex]x=1[\/latex], [latex]z=2[\/latex], and [latex]w=3[\/latex].<\/li>\n
                                                                              7. [latex]y[\/latex] varies jointly as [latex]x[\/latex] and [latex]z[\/latex] and inversely as [latex]w[\/latex]. When [latex]x=5[\/latex], [latex]z=2[\/latex], and [latex]w=20[\/latex], then [latex]y=4[\/latex]. Find [latex]y[\/latex] when [latex]x=3[\/latex] and [latex]z=8[\/latex], and [latex]w=48[\/latex].<\/li>\n
                                                                              8. [latex]y[\/latex] varies jointly as the square of [latex]x[\/latex] and the cube of [latex]z[\/latex] and inversely as the square root of [latex]w[\/latex]. When [latex]x=2[\/latex], [latex]z=2[\/latex], and [latex]w=64[\/latex], then [latex]y=12[\/latex]. Find [latex]y[\/latex] when [latex]x=1[\/latex], [latex]z=3[\/latex], and [latex]w=4[\/latex].<\/li>\n<\/ol>\n

                                                                                For the following exercises, use the given information to answer the questions.<\/p>\n

                                                                                  \n
                                                                                1. The distance [latex]s[\/latex] that an object falls varies directly with the square of the time, [latex]t[\/latex], of the fall. If an object falls [latex]16[\/latex] feet in [latex]1[\/latex] second, how long for it to fall [latex]144[\/latex] feet?<\/li>\n
                                                                                2. The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is [latex]24[\/latex] inches long and vibrates [latex]128[\/latex] times per second, what is the length of a string that vibrates [latex]64[\/latex] times per second?<\/li>\n
                                                                                3. The weight of an object above the surface of Earth varies inversely with the square of the distance from the center of Earth. If a body weighs [latex]50[\/latex] pounds when it is [latex]3960[\/latex] miles from Earth’s center, what would it weigh if it were [latex]3970[\/latex] miles from Earth’s center?<\/li>\n
                                                                                4. The current in a circuit varies inversely with its resistance measured in ohms. When the current in a circuit is [latex]40[\/latex] amperes, the resistance is [latex]10[\/latex] ohms. Find the current if the resistance is [latex]12[\/latex] ohms.<\/li>\n
                                                                                5. The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm) and the cube of the diameter. If the shaft of a certain material [latex]3[\/latex] inches in diameter can transmit [latex]45[\/latex] hp at [latex]100[\/latex] rpm, what must the diameter be in order to transmit [latex]60[\/latex] hp at [latex]150[\/latex] rpm?<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":22,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":232,"module-header":"practice","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4522"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":4,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4522\/revisions"}],"predecessor-version":[{"id":6010,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4522\/revisions\/6010"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/232"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/4522\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=4522"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=4522"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=4522"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=4522"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}