{"id":3812,"date":"2024-09-10T14:00:29","date_gmt":"2024-09-10T14:00:29","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=3812"},"modified":"2024-11-21T16:57:01","modified_gmt":"2024-11-21T16:57:01","slug":"function-basic-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/function-basic-get-stronger\/","title":{"raw":"Function Basic: Get Stronger","rendered":"Function Basic: Get Stronger"},"content":{"raw":"

Introduction to Functions<\/span><\/h2>\r\nFor the following exercises, determine whether the relation represents [latex]y[\/latex] as a function of [latex]x[\/latex].\r\n
    \r\n \t
  1. [latex]y = x^2[\/latex]<\/li>\r\n \t
  2. [latex]3 - \\sqrt{6-2x}[\/latex]<\/li>\r\n \t
  3. [latex]3x^2 + y = 14[\/latex]<\/li>\r\n \t
  4. [latex]y = -2x^2 + 40x[\/latex]<\/li>\r\n \t
  5. [latex]x = \\dfrac{3y + 5}{7y - 1}[\/latex]<\/li>\r\n \t
  6. [latex]y = \\dfrac{3x + 5}{7x - 1}[\/latex]<\/li>\r\n \t
  7. [latex]2xy = 1[\/latex]<\/li>\r\n \t
  8. [latex]y = x^3[\/latex]<\/li>\r\n \t
  9. [latex]x = \\pm \\sqrt{1 - y}[\/latex]<\/li>\r\n \t
  10. [latex]y^2 = x^2[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, evaluate [latex]f(-3)[\/latex], [latex]f(2)[\/latex], [latex]f(-a)[\/latex], [latex]-f(a)[\/latex], [latex]f(a + h)[\/latex].\r\n
      \r\n \t
    1. [latex]f(x) = 2x - 5[\/latex]<\/li>\r\n \t
    2. [latex]f(x) = \\sqrt{2 - x + 5}[\/latex]<\/li>\r\n \t
    3. [latex]f(x) = |x - 1| - |x + 1|[\/latex]<\/li>\r\n \t
    4. Given the function [latex]g(x) = x^2 + 2x[\/latex], evaluate [latex]\\dfrac{g(x) - g(a)}{x - a}[\/latex], [latex]x \\neq a[\/latex].<\/li>\r\n \t
    5. Given the function [latex]f(x) = 8 - 3x[\/latex]:\r\n
        \r\n \t
      1. Evaluate [latex]f(-2)[\/latex].<\/li>\r\n \t
      2. Solve [latex]f(x) = -1[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
      3. Given the function [latex]f(x) = x^2 - 3x[\/latex]:\r\n
          \r\n \t
        1. Evaluate [latex]f(5)[\/latex].<\/li>\r\n \t
        2. Solve [latex]f(x) = 4[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, use the vertical line test to determine which graphs show relations that are functions.\r\n
            \r\n \t
          1. \"\"<\/li>\r\n \t
          2. \"\"<\/li>\r\n \t
          3. \"\"<\/li>\r\n \t
          4. \"\"<\/li>\r\n \t
          5. \"\"<\/li>\r\n \t
          6. \"\"<\/li>\r\n \t
          7. Given the following graph,\r\n\"\"\r\n
              \r\n \t
            1. Evaluate [latex]f(0)[\/latex].<\/li>\r\n \t
            2. Solve for [latex]f(x) = -3[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, determine if the given graph is a one-to-one function.\r\n
                \r\n \t
              1. \"\"<\/li>\r\n \t
              2. \"\"<\/li>\r\n \t
              3. \"\"<\/li>\r\n<\/ol>\r\nFor the following exercise, determine whether the relation represents a function.\r\n
                  \r\n \t
                1. [latex]\\{(3, 4), (4, 5), (5, 6)\\}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, determine if the relation represented in table form represents [latex]y[\/latex] as a function of [latex]x[\/latex].\r\n
                    \r\n \t
                  1. \r\n\r\n\r\n\r\n\r\n
                    [latex]x[\/latex]<\/th>\r\n[latex]5[\/latex]<\/td>\r\n[latex]10[\/latex]<\/td>\r\n[latex]15[\/latex]<\/td>\r\n<\/tr>\r\n
                    [latex]y[\/latex]<\/th>\r\n[latex]3[\/latex]<\/td>\r\n[latex]8[\/latex]<\/td>\r\n[latex]14[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t
                  2. \r\n\r\n\r\n\r\n\r\n
                    [latex]x[\/latex]<\/th>\r\n[latex]5[\/latex]<\/td>\r\n[latex]10[\/latex]<\/td>\r\n[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n
                    [latex]y[\/latex]<\/th>\r\n[latex]3[\/latex]<\/td>\r\n[latex]8[\/latex]<\/td>\r\n[latex]14[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n<\/ol>\r\nFor the following exercises, use the function [latex]f[\/latex] represented in the table below.\r\n\r\n\r\n\r\n\r\n
                    [latex]x[\/latex]<\/th>\r\n[latex]0[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]2[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]4[\/latex]<\/td>\r\n[latex]5[\/latex]<\/td>\r\n[latex]6[\/latex]<\/td>\r\n[latex]7[\/latex]<\/td>\r\n[latex]8[\/latex]<\/td>\r\n[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n
                    [latex]f(x)[\/latex]<\/th>\r\n[latex]74[\/latex]<\/td>\r\n[latex]28[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]53[\/latex]<\/td>\r\n[latex]56[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n[latex]36[\/latex]<\/td>\r\n[latex]45[\/latex]<\/td>\r\n[latex]14[\/latex]<\/td>\r\n[latex]47[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
                      \r\n \t
                    1. Solve [latex]f(x) = 1[\/latex].<\/li>\r\n<\/ol>\r\n
                        \r\n \t
                      1. The number of cubic yards of dirt, [latex]D[\/latex], needed to cover a garden with area [latex]a[\/latex] square feet is given by [latex]D = g(a)[\/latex].\r\n
                          \r\n \t
                        1. A garden with area [latex]5000 \\, \\text{ft}^2[\/latex] requires [latex]50 \\, \\text{yd}^3[\/latex] of dirt. Express this information in terms of the function [latex]g[\/latex].<\/li>\r\n \t
                        2. Explain the meaning of the statement [latex]g(100) = 1[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                        3. Let [latex]h(t)[\/latex] be the height above ground, in feet, of a rocket [latex]t[\/latex] seconds after launching. Explain the meaning of each statement:\r\n
                            \r\n \t
                          1. [latex]h(1) = 200[\/latex]<\/li>\r\n \t
                          2. [latex]h(2) = 350[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n

                            Domain and Range<\/span><\/h2>\r\nFor the following exercises, find the domain of each function using interval notation.\r\n
                              \r\n \t
                            1. [latex]f(x) = 5 - 2x^2[\/latex]<\/li>\r\n \t
                            2. [latex]f(x) = \\sqrt{x^2 + 4}[\/latex]<\/li>\r\n \t
                            3. [latex]f(x) = \\sqrt[3]{x-1}[\/latex]<\/li>\r\n \t
                            4. [latex]f(x) = \\dfrac{3x + 1}{4x + 2}[\/latex]<\/li>\r\n \t
                            5. [latex]f(x) = \\dfrac{x - 3}{x^2 + 9x - 22}[\/latex]<\/li>\r\n \t
                            6. [latex]f(x) = \\dfrac{2x^3 - 250}{x^2 - 2x - 15}[\/latex]<\/li>\r\n \t
                            7. [latex]f(x) = \\dfrac{2x + 1}{\\sqrt{5 - x}}[\/latex]<\/li>\r\n \t
                            8. [latex]f(x) = \\dfrac{\\sqrt{x - 6}}{\\sqrt{x - 4}}[\/latex]<\/li>\r\n \t
                            9. [latex]f(x) = \\dfrac{x^2 - 9x}{x^2 - 81}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, write the domain and range of each function using interval notation.\r\n
                                \r\n \t
                              1. \"\"<\/li>\r\n \t
                              2. \"\"<\/li>\r\n \t
                              3. \"\"<\/li>\r\n \t
                              4. \"\"<\/li>\r\n \t
                              5. \"\"<\/li>\r\n \t
                              6. \"\"<\/li>\r\n<\/ol>\r\nFor the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.\r\n
                                  \r\n \t
                                1. [latex]f(x) = \\begin{cases} 2x - 1 & \\text{if } x < 1 \\\\ 1 + x & \\text{if } x \\geq 1 \\end{cases}[\/latex]<\/li>\r\n \t
                                2. [latex]f(x) = \\begin{cases} 3 & \\text{if } x < 0 \\\\ \\sqrt{x} & \\text{if } x \\geq 0 \\end{cases}[\/latex]<\/li>\r\n \t
                                3. [latex]f(x) = \\begin{cases} x^2 & \\text{if } x < 0 \\\\ x + 2 & \\text{if } x \\geq 0 \\end{cases}[\/latex]<\/li>\r\n \t
                                4. [latex]f(x) = \\begin{cases} |x| & \\text{if } x < 2 \\\\ 1 & \\text{if } x \\geq 2 \\end{cases}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercise, given each function [latex]f[\/latex], evaluate [latex]f(-3)[\/latex], [latex]f(-2)[\/latex], [latex]f(-1)[\/latex], and [latex]f(0)[\/latex].\r\n
                                    \r\n \t
                                  1. [latex]f(x) = \\begin{cases} 1 & \\text{if } x \\leq -3 \\\\ 0 & \\text{if } x > -3 \\end{cases}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercises, given each function [latex]f[\/latex], evaluate [latex]f(-1)[\/latex], [latex]f(0)[\/latex], [latex]f(2)[\/latex], and [latex]f(4)[\/latex].\r\n
                                      \r\n \t
                                    1. [latex]f(x) = \\begin{cases} 7x + 3 & \\text{if } x < 0 \\\\ 7x + 6 & \\text{if } x \\geq 0 \\end{cases}[\/latex]<\/li>\r\n \t
                                    2. [latex]f(x) = \\begin{cases} 5x & \\text{if } x < 0 \\\\ 3 & \\text{if } 0 \\leq x \\leq 3 \\\\ x^2 & \\text{if } x > 3 \\end{cases}[\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercise, write the domain for the piecewise function in interval notation.\r\n
                                        \r\n \t
                                      1. [latex]f(x) = \\begin{cases} x^2 - 2 & \\text{if } x < 1 \\\\ -x^2 + 2 & \\text{if } x \\geq 1 \\end{cases}[\/latex]<\/li>\r\n<\/ol>\r\n

                                        Rates of Change and Behavior of Graphs<\/span><\/h2>\r\nFor the following exercises, find the average rate of change of each function on the interval specified for real numbers [latex]b[\/latex] or [latex]h[\/latex] in simplest form.\r\n
                                          \r\n \t
                                        1. [latex]f(x) = 2x^2 + 1[\/latex] on [latex][x, x + h][\/latex]<\/li>\r\n \t
                                        2. [latex]a(t) = \\dfrac{1}{t + 4}[\/latex] on [latex][9, 9 + h][\/latex]<\/li>\r\n \t
                                        3. [latex]j(x) = 3x^3[\/latex] on [latex][1, 1 + h][\/latex]<\/li>\r\n \t
                                        4. Find [latex]\\dfrac{f(x+h)-f(x)}{h}[\/latex] given [latex]f(x) = 2x^2 - 3x[\/latex] on [latex][x, x + h][\/latex]<\/li>\r\n<\/ol>\r\nFor the following exercise, consider the graph of [latex]f[\/latex] shown in the figure below.\r\n\"\"\r\n
                                            \r\n \t
                                          1. Estimate the average rate of change from [latex]x = 2[\/latex] to [latex]x = 5[\/latex].<\/li>\r\n<\/ol>\r\nFor the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.\r\n
                                              \r\n \t
                                            1. \"\"<\/li>\r\n \t
                                            2. \"\"<\/li>\r\n<\/ol>\r\nFor the following exercises, find the average rate of change of each function on the interval specified.\r\n
                                                \r\n \t
                                              1. [latex]h(x) = 5 - 2x^2[\/latex] on [latex][-2, 4][\/latex]<\/li>\r\n \t
                                              2. [latex]g(x) = 3x^3 - 1[\/latex] on [latex][-3, 3][\/latex]<\/li>\r\n \t
                                              3. [latex]p(t) = \\dfrac{(t^2 - 4)(t + 1)}{t^2 + 3}[\/latex] on [latex][-3, 1][\/latex]<\/li>\r\n<\/ol>\r\nReal-World Applications.\r\n
                                                  \r\n \t
                                                1. A driver of a car stopped at a gas station to fill up their gas tank. They looked at their watch, and the time read exactly 3:40 p.m. At this time, they started pumping gas into the tank. At exactly 3:44, the tank was full and the driver noticed that they had pumped [latex]10.7[\/latex] gallons. What is the average rate of flow of the gasoline into the gas tank?<\/li>\r\n \t
                                                2. The graph below illustrates the decay of a radioactive substance over [latex]t[\/latex] days.\r\n\"\"\r\nUse the graph to estimate the average decay rate from [latex]t = 5[\/latex] to [latex]t = 15[\/latex].<\/li>\r\n<\/ol>","rendered":"

                                                  Introduction to Functions<\/span><\/h2>\n

                                                  For the following exercises, determine whether the relation represents [latex]y[\/latex] as a function of [latex]x[\/latex].<\/p>\n

                                                    \n
                                                  1. [latex]y = x^2[\/latex]<\/li>\n
                                                  2. [latex]3 - \\sqrt{6-2x}[\/latex]<\/li>\n
                                                  3. [latex]3x^2 + y = 14[\/latex]<\/li>\n
                                                  4. [latex]y = -2x^2 + 40x[\/latex]<\/li>\n
                                                  5. [latex]x = \\dfrac{3y + 5}{7y - 1}[\/latex]<\/li>\n
                                                  6. [latex]y = \\dfrac{3x + 5}{7x - 1}[\/latex]<\/li>\n
                                                  7. [latex]2xy = 1[\/latex]<\/li>\n
                                                  8. [latex]y = x^3[\/latex]<\/li>\n
                                                  9. [latex]x = \\pm \\sqrt{1 - y}[\/latex]<\/li>\n
                                                  10. [latex]y^2 = x^2[\/latex]<\/li>\n<\/ol>\n

                                                    For the following exercises, evaluate [latex]f(-3)[\/latex], [latex]f(2)[\/latex], [latex]f(-a)[\/latex], [latex]-f(a)[\/latex], [latex]f(a + h)[\/latex].<\/p>\n

                                                      \n
                                                    1. [latex]f(x) = 2x - 5[\/latex]<\/li>\n
                                                    2. [latex]f(x) = \\sqrt{2 - x + 5}[\/latex]<\/li>\n
                                                    3. [latex]f(x) = |x - 1| - |x + 1|[\/latex]<\/li>\n
                                                    4. Given the function [latex]g(x) = x^2 + 2x[\/latex], evaluate [latex]\\dfrac{g(x) - g(a)}{x - a}[\/latex], [latex]x \\neq a[\/latex].<\/li>\n
                                                    5. Given the function [latex]f(x) = 8 - 3x[\/latex]:\n
                                                        \n
                                                      1. Evaluate [latex]f(-2)[\/latex].<\/li>\n
                                                      2. Solve [latex]f(x) = -1[\/latex].<\/li>\n<\/ol>\n<\/li>\n
                                                      3. Given the function [latex]f(x) = x^2 - 3x[\/latex]:\n
                                                          \n
                                                        1. Evaluate [latex]f(5)[\/latex].<\/li>\n
                                                        2. Solve [latex]f(x) = 4[\/latex].<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                                                          For the following exercises, use the vertical line test to determine which graphs show relations that are functions.<\/p>\n

                                                            \n
                                                          1. \"\"<\/li>\n
                                                          2. \"\"<\/li>\n
                                                          3. \"\"<\/li>\n
                                                          4. \"\"<\/li>\n
                                                          5. \"\"<\/li>\n
                                                          6. \"\"<\/li>\n
                                                          7. Given the following graph,
                                                            \n\"\"<\/p>\n
                                                              \n
                                                            1. Evaluate [latex]f(0)[\/latex].<\/li>\n
                                                            2. Solve for [latex]f(x) = -3[\/latex].<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                                                              For the following exercises, determine if the given graph is a one-to-one function.<\/p>\n

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

                                                                For the following exercise, determine whether the relation represents a function.<\/p>\n

                                                                  \n
                                                                1. [latex]\\{(3, 4), (4, 5), (5, 6)\\}[\/latex]<\/li>\n<\/ol>\n

                                                                  For the following exercises, determine if the relation represented in table form represents [latex]y[\/latex] as a function of [latex]x[\/latex].<\/p>\n

                                                                    \n
                                                                  1. \n\n\n\n\n
                                                                    [latex]x[\/latex]<\/th>\n[latex]5[\/latex]<\/td>\n[latex]10[\/latex]<\/td>\n[latex]15[\/latex]<\/td>\n<\/tr>\n
                                                                    [latex]y[\/latex]<\/th>\n[latex]3[\/latex]<\/td>\n[latex]8[\/latex]<\/td>\n[latex]14[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n
                                                                  2. \n\n\n\n\n
                                                                    [latex]x[\/latex]<\/th>\n[latex]5[\/latex]<\/td>\n[latex]10[\/latex]<\/td>\n[latex]10[\/latex]<\/td>\n<\/tr>\n
                                                                    [latex]y[\/latex]<\/th>\n[latex]3[\/latex]<\/td>\n[latex]8[\/latex]<\/td>\n[latex]14[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ol>\n

                                                                    For the following exercises, use the function [latex]f[\/latex] represented in the table below.<\/p>\n\n\n\n\n
                                                                    [latex]x[\/latex]<\/th>\n[latex]0[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]2[\/latex]<\/td>\n[latex]3[\/latex]<\/td>\n[latex]4[\/latex]<\/td>\n[latex]5[\/latex]<\/td>\n[latex]6[\/latex]<\/td>\n[latex]7[\/latex]<\/td>\n[latex]8[\/latex]<\/td>\n[latex]9[\/latex]<\/td>\n<\/tr>\n
                                                                    [latex]f(x)[\/latex]<\/th>\n[latex]74[\/latex]<\/td>\n[latex]28[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]53[\/latex]<\/td>\n[latex]56[\/latex]<\/td>\n[latex]3[\/latex]<\/td>\n[latex]36[\/latex]<\/td>\n[latex]45[\/latex]<\/td>\n[latex]14[\/latex]<\/td>\n[latex]47[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                                                                      \n
                                                                    1. Solve [latex]f(x) = 1[\/latex].<\/li>\n<\/ol>\n
                                                                        \n
                                                                      1. The number of cubic yards of dirt, [latex]D[\/latex], needed to cover a garden with area [latex]a[\/latex] square feet is given by [latex]D = g(a)[\/latex].\n
                                                                          \n
                                                                        1. A garden with area [latex]5000 \\, \\text{ft}^2[\/latex] requires [latex]50 \\, \\text{yd}^3[\/latex] of dirt. Express this information in terms of the function [latex]g[\/latex].<\/li>\n
                                                                        2. Explain the meaning of the statement [latex]g(100) = 1[\/latex].<\/li>\n<\/ol>\n<\/li>\n
                                                                        3. Let [latex]h(t)[\/latex] be the height above ground, in feet, of a rocket [latex]t[\/latex] seconds after launching. Explain the meaning of each statement:\n
                                                                            \n
                                                                          1. [latex]h(1) = 200[\/latex]<\/li>\n
                                                                          2. [latex]h(2) = 350[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                                                                            Domain and Range<\/span><\/h2>\n

                                                                            For the following exercises, find the domain of each function using interval notation.<\/p>\n

                                                                              \n
                                                                            1. [latex]f(x) = 5 - 2x^2[\/latex]<\/li>\n
                                                                            2. [latex]f(x) = \\sqrt{x^2 + 4}[\/latex]<\/li>\n
                                                                            3. [latex]f(x) = \\sqrt[3]{x-1}[\/latex]<\/li>\n
                                                                            4. [latex]f(x) = \\dfrac{3x + 1}{4x + 2}[\/latex]<\/li>\n
                                                                            5. [latex]f(x) = \\dfrac{x - 3}{x^2 + 9x - 22}[\/latex]<\/li>\n
                                                                            6. [latex]f(x) = \\dfrac{2x^3 - 250}{x^2 - 2x - 15}[\/latex]<\/li>\n
                                                                            7. [latex]f(x) = \\dfrac{2x + 1}{\\sqrt{5 - x}}[\/latex]<\/li>\n
                                                                            8. [latex]f(x) = \\dfrac{\\sqrt{x - 6}}{\\sqrt{x - 4}}[\/latex]<\/li>\n
                                                                            9. [latex]f(x) = \\dfrac{x^2 - 9x}{x^2 - 81}[\/latex]<\/li>\n<\/ol>\n

                                                                              For the following exercises, write the domain and range of each function using interval notation.<\/p>\n

                                                                                \n
                                                                              1. \"\"<\/li>\n
                                                                              2. \"\"<\/li>\n
                                                                              3. \"\"<\/li>\n
                                                                              4. \"\"<\/li>\n
                                                                              5. \"\"<\/li>\n
                                                                              6. \"\"<\/li>\n<\/ol>\n

                                                                                For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.<\/p>\n

                                                                                  \n
                                                                                1. [latex]f(x) = \\begin{cases} 2x - 1 & \\text{if } x < 1 \\\\ 1 + x & \\text{if } x \\geq 1 \\end{cases}[\/latex]<\/li>\n
                                                                                2. [latex]f(x) = \\begin{cases} 3 & \\text{if } x < 0 \\\\ \\sqrt{x} & \\text{if } x \\geq 0 \\end{cases}[\/latex]<\/li>\n
                                                                                3. [latex]f(x) = \\begin{cases} x^2 & \\text{if } x < 0 \\\\ x + 2 & \\text{if } x \\geq 0 \\end{cases}[\/latex]<\/li>\n
                                                                                4. [latex]f(x) = \\begin{cases} |x| & \\text{if } x < 2 \\\\ 1 & \\text{if } x \\geq 2 \\end{cases}[\/latex]<\/li>\n<\/ol>\n

                                                                                  For the following exercise, given each function [latex]f[\/latex], evaluate [latex]f(-3)[\/latex], [latex]f(-2)[\/latex], [latex]f(-1)[\/latex], and [latex]f(0)[\/latex].<\/p>\n

                                                                                    \n
                                                                                  1. [latex]f(x) = \\begin{cases} 1 & \\text{if } x \\leq -3 \\\\ 0 & \\text{if } x > -3 \\end{cases}[\/latex]<\/li>\n<\/ol>\n

                                                                                    For the following exercises, given each function [latex]f[\/latex], evaluate [latex]f(-1)[\/latex], [latex]f(0)[\/latex], [latex]f(2)[\/latex], and [latex]f(4)[\/latex].<\/p>\n

                                                                                      \n
                                                                                    1. [latex]f(x) = \\begin{cases} 7x + 3 & \\text{if } x < 0 \\\\ 7x + 6 & \\text{if } x \\geq 0 \\end{cases}[\/latex]<\/li>\n
                                                                                    2. [latex]f(x) = \\begin{cases} 5x & \\text{if } x < 0 \\\\ 3 & \\text{if } 0 \\leq x \\leq 3 \\\\ x^2 & \\text{if } x > 3 \\end{cases}[\/latex]<\/li>\n<\/ol>\n

                                                                                      For the following exercise, write the domain for the piecewise function in interval notation.<\/p>\n

                                                                                        \n
                                                                                      1. [latex]f(x) = \\begin{cases} x^2 - 2 & \\text{if } x < 1 \\\\ -x^2 + 2 & \\text{if } x \\geq 1 \\end{cases}[\/latex]<\/li>\n<\/ol>\n

                                                                                        Rates of Change and Behavior of Graphs<\/span><\/h2>\n

                                                                                        For the following exercises, find the average rate of change of each function on the interval specified for real numbers [latex]b[\/latex] or [latex]h[\/latex] in simplest form.<\/p>\n

                                                                                          \n
                                                                                        1. [latex]f(x) = 2x^2 + 1[\/latex] on [latex][x, x + h][\/latex]<\/li>\n
                                                                                        2. [latex]a(t) = \\dfrac{1}{t + 4}[\/latex] on [latex][9, 9 + h][\/latex]<\/li>\n
                                                                                        3. [latex]j(x) = 3x^3[\/latex] on [latex][1, 1 + h][\/latex]<\/li>\n
                                                                                        4. Find [latex]\\dfrac{f(x+h)-f(x)}{h}[\/latex] given [latex]f(x) = 2x^2 - 3x[\/latex] on [latex][x, x + h][\/latex]<\/li>\n<\/ol>\n

                                                                                          For the following exercise, consider the graph of [latex]f[\/latex] shown in the figure below.
                                                                                          \n\"\"<\/p>\n

                                                                                            \n
                                                                                          1. Estimate the average rate of change from [latex]x = 2[\/latex] to [latex]x = 5[\/latex].<\/li>\n<\/ol>\n

                                                                                            For the following exercises, use the graph of each function to estimate the intervals on which the function is increasing or decreasing.<\/p>\n

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

                                                                                              For the following exercises, find the average rate of change of each function on the interval specified.<\/p>\n

                                                                                                \n
                                                                                              1. [latex]h(x) = 5 - 2x^2[\/latex] on [latex][-2, 4][\/latex]<\/li>\n
                                                                                              2. [latex]g(x) = 3x^3 - 1[\/latex] on [latex][-3, 3][\/latex]<\/li>\n
                                                                                              3. [latex]p(t) = \\dfrac{(t^2 - 4)(t + 1)}{t^2 + 3}[\/latex] on [latex][-3, 1][\/latex]<\/li>\n<\/ol>\n

                                                                                                Real-World Applications.<\/p>\n

                                                                                                  \n
                                                                                                1. A driver of a car stopped at a gas station to fill up their gas tank. They looked at their watch, and the time read exactly 3:40 p.m. At this time, they started pumping gas into the tank. At exactly 3:44, the tank was full and the driver noticed that they had pumped [latex]10.7[\/latex] gallons. What is the average rate of flow of the gasoline into the gas tank?<\/li>\n
                                                                                                2. The graph below illustrates the decay of a radioactive substance over [latex]t[\/latex] days.
                                                                                                  \n\"\"
                                                                                                  \nUse the graph to estimate the average decay rate from [latex]t = 5[\/latex] to [latex]t = 15[\/latex].<\/li>\n<\/ol>\n","protected":false},"author":15,"menu_order":26,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":116,"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\/3812"}],"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":13,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3812\/revisions"}],"predecessor-version":[{"id":6256,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3812\/revisions\/6256"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/116"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/3812\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=3812"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=3812"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=3812"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=3812"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}