Basic Functions and Graphs: Get Stronger Answer Key

Review of Functions

    1. Domain = [latex]\{-3,-2,-1,0,1,2,3\}[/latex], range = [latex]\{0,1,4,9\}[/latex]
    2. Yes, a function
    1. Domain = [latex]\{0,1,2,3\}[/latex], range = [latex]\{-3,-2,-1,0,1,2,3\}[/latex]
    2. No, not a function
    1. Domain = [latex]\{3,5,8,10,15,21,33\}[/latex], range = [latex]\{0,1,2,3\}[/latex]
    2. Yes, a function
    1. [latex]-2[/latex]
    2. [latex]3[/latex]
    3. [latex]13[/latex]
    4. [latex]-5x-2[/latex]
    5. [latex]5a-2[/latex]
    6. [latex]5a+5h-2[/latex]
    1. Undefined
    2. [latex]2[/latex]
    3. [latex]\frac{2}{3}[/latex]
    4. [latex]-\frac{2}{x}[/latex]
    5. [latex]\frac{2}{a}[/latex]
    6. [latex]\frac{2}{a+h}[/latex]
    1. [latex]\sqrt{5}[/latex]
    2. [latex]\sqrt{11}[/latex]
    3. [latex]\sqrt{23}[/latex]
    4. [latex]\sqrt{-6x+5}[/latex]
    5. [latex]\sqrt{6a+5}[/latex]
    6. [latex]\sqrt{6a+6h+5}[/latex]
    1. [latex]9[/latex]
    2. [latex]9[/latex]
    3. [latex]9[/latex]
    4. [latex]9[/latex]
    5. [latex]9[/latex]
  8. Domain: [latex]x\ge \frac{1}{8}[/latex]; Range: [latex]y\ge 0[/latex]; Zeros: [latex]x=\frac{1}{8}[/latex]; no [latex]y[/latex]-intercepts
  9. Domain: [latex]x\ge -2[/latex]; Range: [latex]y\ge -1[/latex]; Zeros: [latex]x=-1[/latex]; [latex]y[/latex]-intercepts: [latex]y=-1+\sqrt{2}[/latex]
  10. Domain: [latex]x\ne 4[/latex]; Range: [latex]y\ne 0[/latex]; no [latex]x[/latex]-intercept; [latex]y[/latex]-intercept: [latex]y=-\frac{3}{4}[/latex]
  11. Domain: [latex]x>5[/latex]; Range: [latex]y>0[/latex]; no intercepts
  12. [latex]x[/latex] [latex]y[/latex] [latex]x[/latex] [latex]y[/latex]
    [latex]−3[/latex] [latex]−15[/latex] [latex]1[/latex] [latex]−3[/latex]
    [latex]−2[/latex] [latex]−12[/latex] [latex]2[/latex] [latex]0[/latex]
    [latex]−1[/latex] [latex]−9[/latex] [latex]3[/latex] [latex]3[/latex]
    [latex]0[/latex] [latex]−6[/latex]    

    An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -3 to 3. The graph is of the function “f(x) = 3x - 6”, which is an increasing straight line. The function has an x intercept at (2, 0) and the y intercept is not shown.

  13. [latex]x[/latex] [latex]y[/latex] [latex]x[/latex] [latex]y[/latex]
    [latex]−3[/latex] [latex]6[/latex] [latex]1[/latex] [latex]2[/latex]
    [latex]−2[/latex] [latex]4[/latex] [latex]2[/latex] [latex]4[/latex]
    [latex]−1[/latex] [latex]2[/latex] [latex]3[/latex] [latex]6[/latex]
    [latex]0[/latex] [latex]0[/latex]    

    An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -2 to 6. The graph is of the function “f(x) = 2 times the absolute value of x”. The function decreases in a straight line until it hits the origin, then begins to increase in a straight line. The function x intercept and y intercept are at the origin.

  14. [latex]x[/latex] [latex]y[/latex] [latex]x[/latex] [latex]y[/latex]
    [latex]−3[/latex] [latex]−27[/latex] [latex]1[/latex] [latex]1[/latex]
    [latex]−2[/latex] [latex]−8[/latex] [latex]2[/latex] [latex]8[/latex]
    [latex]−1[/latex] [latex]−1[/latex] [latex]3[/latex] [latex]27[/latex]
    [latex]0[/latex] [latex]0[/latex]    

    An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -27 to 27. The graph is of the function “f(x) = x cubed”. The curved function increases until it hits the origin, where it levels out and then becomes even. After the origin the graph begins to increase again. The x intercept and y intercept are both at the origin.

  15. Function
    1. Domain: all real numbers, Range: [latex]y\ge 0[/latex]
    2. [latex]x=±1[/latex]
    3. [latex]y=1[/latex]
    4. [latex]-1 < x< 0[/latex] and [latex]1 < x < \infty[/latex]
    5. [latex]−\infty < x < -1[/latex] and [latex]0 < x < 1[/latex]
    6. Not constant
    7. [latex]y[/latex]-axis
    8. Even
  16. Function
    1. Domain: all real numbers, Range: [latex]-1.5\le y\le 1.5[/latex]
    2. [latex]x=0[/latex]
    3. [latex]y=0[/latex]
    4. All real numbers
    5. None
    6. Not constant
    7. Origin
    8. Odd
  17. Function
    1. Domain: [latex]−\infty > x > \infty[/latex], Range: [latex]-2\le y\le 2[/latex]
    2. [latex]x=0[/latex]
    3. [latex]y=0[/latex]
    4. [latex]-2 > x > 2[/latex]
    5. Not decreasing
    6. [latex]−\infty > x > -2[/latex] and [latex]2 > x > \infty[/latex]
    7. Origin
    8. Odd
  18. Function
    1. Domain: [latex]-4\le x\le 4[/latex], Range: [latex]-4\le y\le 4[/latex]
    2. [latex]x=1.2[/latex]
    3. [latex]y=4[/latex]
    4. Not increasing
    5. [latex]0 > x > 4[/latex]
    6. [latex]-4 > x > 0[/latex]
    7. No Symmetry
    8. Neither
    1. [latex]5x^2+x-8[/latex]; all real numbers
    2. [latex]-5x^2+x-8[/latex]; all real numbers
    3. [latex]5x^3-40x^2[/latex]; all real numbers
    4. [latex]\frac{x-8}{5x^2}; \, x\ne 0[/latex]
    1. [latex]-2x+6[/latex]; all real numbers
    2. [latex]-2x^2+2x+12[/latex]; all real numbers
    3. [latex]−x^4+2x^3+12x^2-18x-27[/latex]; all real numbers
    4. [latex]-\frac{x+3}{x+1}; \, x\ne −1,3[/latex]
    1. [latex]6+\frac{2}{x}; \, x\ne 0[/latex]
    2. 6; [latex]x\ne 0[/latex]
    3. [latex]\frac{6}{x}+\frac{1}{x^2}; \, x\ne 0[/latex]
    4. [latex]6x+1; \, x\ne 0[/latex]
    1. [latex]4x+3[/latex]; all real numbers
    2. [latex]4x+15[/latex]; all real numbers
    1. [latex]x^4-6x^2+16[/latex]; all real numbers
    2. [latex]x^4+14x^2+46[/latex]; all real numbers
    1. [latex]\frac{3x}{4+x}; \, x\ne 0,-4[/latex]
    2. [latex]\frac{4x+2}{3}; \, x\ne -\frac{1}{2}[/latex]
    1. Yes, because there is only one winner for each year.
    2. No, because there are three teams that won more than once during the years 2001 to 2012.
    1. [latex]V(s)=s^3[/latex]
    2. [latex]V(11.8)\approx 1643[/latex]; a cube of side length [latex]11.8[/latex] each has a volume of approximately [latex]1643[/latex] cubic units.
    1. [latex]N(x)=15x[/latex]
    2. i. [latex]N(20)=15(20)=300[/latex]; therefore, the vehicle can travel [latex]300[/latex] mi on a full tank of gas. ii. [latex]N(15)=225[/latex]; therefore, the vehicle can travel [latex]225[/latex] mi on [latex]3/4[/latex] of a tank of gas.
    3. Domain: [latex]0\le x\le 20[/latex]; Range: [latex][0,300][/latex]
    4. The driver had to stop at least once, given that it takes approximately [latex]39[/latex] gal of gas to drive a total of [latex]578[/latex] mi.
    1. [latex]A(t)=A(r(t))=\pi ·(6-\frac{5}{t^2+1})^2[/latex]
    2. Exact: [latex]\frac{121\pi }{4}[/latex]; approximately [latex]95[/latex] cm2
    3. [latex]C(t)=C(r(t))=2\pi (6-\frac{5}{t^2+1})[/latex]
    4. Exact: [latex]11\pi[/latex]; approximately [latex]35[/latex] cm
  29. An image of a graph. The y axis runs from 0 to 1800 and the x axis runs from 0 to 100. The graph is of the function “S(x) = 8.5x + 750”, which is a increasing straight line. The function has a y intercept at (0, 750) and the x intercept is not shown.
    1. [latex]S(x)=8.5x+750[/latex]
    2. [latex]$962.50[/latex], [latex]$1090[/latex], [latex]$1217.50[/latex]
    3. [latex]77[/latex] skateboards

Basic Classes of Functions

      1. [latex]−1[/latex]
      2. Decreasing
      1. [latex]3/4[/latex]
      2. Increasing
      1. [latex]4/3[/latex]
      2. Increasing
      1. [latex]0[/latex]
      2. Horizontal
    5. [latex]y=-6x+9[/latex]
    6. [latex]y=\frac{1}{3}x+4[/latex]
    7. [latex]y=\frac{1}{2}x[/latex]
    8. [latex]y=\frac{3}{5}x-3[/latex]
      1. [latex](m=2, \, b=-3)[/latex]
      2. An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph shows an increasing straight line function with a y intercept at (0, -3) and a x intercept at (1.5, 0).
      1. [latex](m=-6, \, b=0)[/latex]
      2. An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph shows a decreasing straight line function with a y intercept and x intercept both at the origin. There is an unlabeled point on the function at (0.5, -3).
      1. [latex](m=0, \, b=-6)[/latex]
      2. An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -7 to 1. The graph shows a horizontal straight line function with a y intercept at (0, -6) and no x intercept.
      1. [latex](m=-\frac{2}{3}, \, b=2)[/latex]
      2. An image of a graph. The x axis runs from -3 to 3 and the y axis runs from -4 to 4. The graph shows a decreasing straight line function with a y intercept at (0, 2) and a x intercept at (3, 0).
      1. [latex]2[/latex]
      2. [latex]\frac{5}{2}, \, -1[/latex]
      3. [latex]−5[/latex]
      4. Both ends rise
      5. Neither
      1. [latex]2[/latex]
      2. [latex]\pm \sqrt{2}[/latex]
      3. [latex]−1[/latex]
      4. Both ends rise
      5. Even
      1. [latex]3[/latex]
      2. [latex]0[/latex], [latex]\pm \sqrt{3}[/latex]
      3. [latex]0[/latex]
      4. Left end rises, right end falls
      5. Odd
    16. An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph shows a parabolic function that decreases until the point (-3, 1), then begins increasing. The y intercept is not shown and there are no x intercepts. There are two unplotted points at (-4, 2) and (-2, 2).
    17. An image of a graph. The x axis runs from -5 to 20 and the y axis runs from -8 to 2. The graph shows a curved function that begins at the point (0, -1), then begins decreasing. The y intercept is at (0, -1) and there is no x intercept. There is an unplotted point at (9, -4).
    18. An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph shows a function that starts at point (-2, 2), where it begins to increase until the point (0, 4). After the point (0, 4), the function becomes a horizontal line and stays that way until the point (2, 4). After the point (2, 4), the function begins to decrease until the point (4, 2), where the function ends.
      1. [latex]13, \, -3, \, 5[/latex]
      2. An image of a graph. The x axis runs from -5 to 5 and the y axis runs from -5 to 5. The graph is of a function that has two pieces. The first piece is a decreasing curve that ends at the point (0, -3). The second piece is an increasing line that begins at the point (0, -3). The function has a x intercepts at the approximate point (1.7, 0) and the point (0.75, 0) and a y intercept at (0, -3).
      1. [latex]\frac{-3}{2}, \, \frac{-1}{2}, \, 4[/latex]
      2. An image of a graph. The x axis runs from -10 to 10 and the y axis runs from -10 to 10. The graph is of a function that begins slightly below the x axis and begins to decrease. As the function approaches the unplotted vertical line of “x = 2”, it decreases at a faster rate but never reaches the line “x = 2”. On the right side of the unplotted line “x = 2”, the function starts at the top of graph and begins decreasing and approaches the unplotted horizontal line “y = 0”, but never reaches “y = 0”. There function also includes a plotted point at (2, 4). There is a y intercept at (0, -1.5) and no x intercept.
    21. True, because [latex]n=3[/latex]
    22. False, because [latex]f(x)=x^b[/latex] – where [latex]b[/latex] is a real-valued constant – is a power function.  Exponential functions are of the form [latex]f(x)=b^x[/latex], where [latex]b[/latex] is a real-valued constant.
      1. [latex]V(t)=-2733t+20500[/latex]
      2. [latex](0, 20500)[/latex] means that the initial purchase price of the equipment is [latex]$20,500[/latex]; [latex](7.5,0)[/latex] means that in [latex]7.5[/latex] years the computer equipment has no value.
      3. [latex]$6835[/latex]
      4. In approximately [latex]6.4[/latex] years
      1. [latex]C=0.75x+125[/latex]
      2. [/latex]$245[latex]
      3. [latex]167[/latex] cupcakes
      1. [latex]V(t)=-1500t+26,000[/latex]
      2. In [latex]4[/latex] years, the value of the car is [latex]$20,000[/latex].
    26. [latex]$30,337.50[/latex]
    27. [latex]96\%[/latex] of the total capacity