Basic Functions and Graphs: Get Stronger

Review of Functions

For the following exercises (1-3),

  1. determine the domain and the range of each relation
  2. state whether the relation is a function.
  1. [latex]x[/latex] [latex]y[/latex] [latex]x[/latex] [latex]y[/latex]
    [latex]−3[/latex] [latex]9[/latex] [latex]1[/latex] [latex]1[/latex]
    [latex]−2[/latex] [latex]4[/latex] [latex]2[/latex] [latex]4[/latex]
    [latex]−1[/latex] [latex]1[/latex] [latex]3[/latex] [latex]9[/latex]
    [latex]0[/latex] [latex]0[/latex]    
  2. [latex]x[/latex] [latex]y[/latex] [latex]x[/latex] [latex]y[/latex]
    [latex]1[/latex] [latex]−3[/latex] [latex]1[/latex] [latex]1[/latex]
    [latex]2[/latex] [latex]−2[/latex] [latex]2[/latex] [latex]2[/latex]
    [latex]3[/latex] [latex]−1[/latex] [latex]3[/latex] [latex]3[/latex]
    [latex]0[/latex] [latex]0[/latex]    
  3. [latex]x[/latex] [latex]y[/latex] [latex]x[/latex] [latex]y[/latex]
    [latex]3[/latex] [latex]3[/latex] [latex]15[/latex] [latex]1[/latex]
    [latex]5[/latex] [latex]2[/latex] [latex]21[/latex] [latex]2[/latex]
    [latex]8[/latex] [latex]1[/latex] [latex]33[/latex] [latex]3[/latex]
    [latex]10[/latex] [latex]0[/latex]    

For the following exercises (4-7), find the values for each of the following functions (a-f), if they exist, then simplify.

  1. [latex]f(0)[/latex]
  2. [latex]f(1)[/latex]
  3. [latex]f(3)[/latex]
  4. [latex]f(−x)[/latex]
  5. [latex]f(a)[/latex]
  6. [latex]f(a+h)[/latex]
  1. [latex]f(x)=5x-2[/latex]
  2. [latex]f(x)=\dfrac{2}{x}[/latex]
  3. [latex]f(x)=\sqrt{6x+5}[/latex]
  4. [latex]f(x)=9[/latex]

For the following exercises (8-11), find the,

  1. domain of the functions
  2. range of the functions
  3. all zeros/intercepts, if any, of the functions
  1. [latex]g(x)=\sqrt{8x-1}[/latex]
  2. [latex]f(x)=-1+\sqrt{x+2}[/latex]
  3. [latex]g(x)=\dfrac{3}{x-4}[/latex]
  4. [latex]g(x)=\sqrt{\dfrac{7}{x-5}}[/latex]

For the following exercises (12-14), use a table of values to sketch the graph of each function using the following values:

[latex]x=-3,-2,-1,0,1,2,3[/latex]

  1. [latex]f(x)=3x-6[/latex]
  2. [latex]f(x)=2|x|[/latex]
  3. [latex]f(x)=x^3[/latex]

For the following exercises (15-18), use the vertical line test to determine whether each of the given graphs represents a function. Assume that a graph continues at both ends if it extends beyond the given grid. If the graph represents a function, then determine the following for each graph:

  1. Domain and range
  2. [latex]x[/latex]-intercept, if any (estimate where necessary)
  3. [latex]y[/latex]-intercept, if any (estimate where necessary)
  4. The intervals for which the function is increasing
  5. The intervals for which the function is decreasing
  6. The intervals for which the function is constant
  7. Symmetry about any axis and/or the origin
  8. Whether the function is even, odd, or neither
  1. 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 relation that is curved. The relation decreases until it hits the point (-1, 0), then increases until it hits the point (0, 1), then decreases until it hits the point (1, 0), then increases again.
  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 relation that is curved. The curved relation increases the entire time. The x intercept and y intercept are both at the origin.
  3. 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 relation that is a horizontal line until the point (-2, -2), then it begins increasing in a straight line until the point (2, 2). After these points, the relation becomes a horizontal line again. The x intercept and y intercept are both at the origin.
  4. 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 relation that starts at the point (-4, 4) and is a horizontal line until the point (0, 4), then it begins decreasing in a curved line until it hits the point (4, -4), where the graph ends. The x intercept is approximately at the point (1.2, 0) and y intercept is at the point (0, 4).

For the following exercises (19-21), for each pair of functions, find

  1. [latex]f+g[/latex]
  2. [latex]f-g[/latex]
  3. [latex]f·g[/latex]
  4. [latex]f/g[/latex]

Determine the domain of each of these new functions.

  1. [latex]f(x)=x-8,\,\,\, g(x)=5x^2[/latex]
  2. [latex]f(x)=9-x^2,\,\,\, g(x)=x^2-2x-3[/latex]
  3. [latex]f(x)=6+\dfrac{1}{x},\,\,\, g(x)=\dfrac{1}{x}[/latex]

For the following exercises (22-24), for each pair of functions, find

  1. [latex](f\circ g)(x)[/latex]
  2. [latex](g\circ f)(x)[/latex]

Simplify the results. Find the domain of each of the results.

  1. [latex]f(x)=x+4,\,\,\, g(x)=4x-1[/latex]
  2. [latex]f(x)=x^2+7,\,\,\, g(x)=x^2-3[/latex]
  3. [latex]f(x)=\dfrac{3}{2x+1},\,\,\, g(x)=\dfrac{2}{x}[/latex]

For the following exercises (25-29), complete all parts of each question. You may utilize calculators or any technological aids to facilitate your calculations.

  1. The table below lists the NBA championship winners for the years 2001 to 2012.
    Year Winner
    2001 LA Lakers
    2002 LA Lakers
    2003 San Antonio Spurs
    2004 Detroit Pistons
    2005 San Antonio Spurs
    2006 Miami Heat
    2007 San Antonio Spurs
    2008 Boston Celtics
    2009 LA Lakers
    2010 LA Lakers
    2011 Dallas Mavericks
    2012 Miami Heat
    1. Consider the relation in which the domain values are the years 2001 to 2012 and the range is the corresponding winner. Is this relation a function? Explain why or why not.
    2. Consider the relation where the domain values are the winners and the range is the corresponding years. Is this relation a function? Explain why or why not.
  2. The volume of a cube depends on the length of the sides [latex]s[/latex].
    1. Write a function [latex]V(s)[/latex] for the area of a square.
    2. Find and interpret [latex]V(11.8)[/latex].
  3. A vehicle has a [latex]20[/latex]-gal tank and gets [latex]15[/latex] mpg. The number of miles [latex]N[/latex] that can be driven depends on the amount of gas [latex]x[/latex] in the tank.
    1. Write a formula that models this situation.
    2. Determine the number of miles the vehicle can travel on (i) a full tank of gas and (ii) [latex]3/4[/latex] of a tank of gas.
    3. Determine the domain and range of the function.
    4. Determine how many times the driver had to stop for gas if she has driven a total of [latex]578[/latex] mi.
  4. A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by [latex]r(t)=6-\left[\dfrac{5}{(t^2+1)}\right][/latex], where [latex]t[/latex] is time measured in hours since a circle of a [latex]1[/latex] cm radius of the bacterium was put into the culture.
    1. Express the area of the bacteria as a function of time.
    2. Find the exact and approximate area of the bacterial culture in [latex]3[/latex] hours.
    3. Express the circumference of the bacteria as a function of time.
    4. Find the exact and approximate circumference of the bacteria in [latex]3[/latex] hours.
  5. The manager at a skateboard shop pays his workers a monthly salary [latex]S[/latex] of [latex]$750[/latex] plus a commission of [latex]$8.50[/latex] for each skateboard they sell.
    1. Write a function [latex]y=S(x)[/latex] that models a worker’s monthly salary based on the number of skateboards [latex]x[/latex] he or she sells.
    2. Find the approximate monthly salary when a worker sells [latex]25[/latex], [latex]40[/latex], or [latex]55[/latex] skateboards.
    3. Use the INTERSECT feature on a graphing calculator to determine the number of skateboards that must be sold for a worker to earn a monthly income of [latex]$1400[/latex].

Basic Classes of Functions

For the following exercises (1-4), for each pair of points,

  1. find the slope of the line passing through the points
  2. indicate whether the line is increasing, decreasing, horizontal, or vertical
  1. [latex](-2,4)[/latex] and [latex](1,1)[/latex]
  2. [latex](3,5)[/latex] and [latex](-1,2)[/latex]
  3. [latex](2,3)[/latex] and [latex](5,7)[/latex]
  4. [latex](2,4)[/latex] and [latex](1,4)[/latex]

For the following exercises (5-8), write the equation of the line satisfying the given conditions in slope-intercept form.

  1. Slope [latex]=-6[/latex], passes through [latex](1,3)[/latex]
  2. Slope [latex]=\dfrac{1}{3}[/latex], passes through [latex](0,4)[/latex]
  3. Passing through [latex](2,1)[/latex] and [latex](-2,-1)[/latex]
  4. [latex]x[/latex]-intercept [latex]=5[/latex] and [latex]y[/latex]-intercept [latex]=-3[/latex]

For the following exercises (9-12), for each linear equation,

  1. give the slope [latex](m)[/latex], and [latex]y[/latex]-intercept [latex](b)[/latex], if any
  2. graph the line
  1. [latex]y=2x-3[/latex]
  2. [latex]f(x)=-6x[/latex]
  3. [latex]4y+24=0[/latex]
  4. [latex]2x+3y=6[/latex]

For the following exercises (13-15), for each polynomial,

  1. find the degree
  2. find the zeros, if any
  3. find the [latex]y[/latex]-intercept(s), if any
  4. use the leading coefficient to determine the graph’s end behavior
  5. determine algebraically whether the polynomial is even, odd, or neither.
  1. [latex]f(x)=2x^2-3x-5[/latex]
  2. [latex]f(x)=\frac{1}{2}x^2-1[/latex]
  3. [latex]f(x)=3x-x^3[/latex]

For the following exercise (16), use the graph of [latex]f(x)=x^2[/latex] to graph the transformed function [latex]g[/latex].

  1. [latex]g(x)=(x+3)^2+1[/latex]

For the following exercise (17), use the graph of [latex]f(x)=\sqrt{x}[/latex] to graph the transformed function [latex]g[/latex].

  1. [latex]g(x)=−\sqrt{x}-1[/latex]

For the following exercise (18), use the graph of [latex]y=f(x)[/latex] to graph the transformed function [latex]g[/latex].

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 (-3, 0), where it begins to increase until the point (-1, 2). After the point (-1, 2), the function becomes a horizontal line and stays that way until the point (1, 2). After the point (1, 2), the function begins to decrease until the point (3, 0), where the function ends.

  1. [latex]g(x)=f(x-1)+2[/latex]

For the following exercises (19-20), for each of the piecewise-defined functions,

  1. Evaluate at the given values of the independent variable
  2. Sketch the graph
  1. [latex]f(x)=\begin{cases}x^2-3, & x < 0 \\ 4x-3, & x \ge 0 \end{cases}[/latex];   [latex]f(-4); \, f(0); \, f(2)[/latex]
  2. [latex]g(x)=\begin{cases} \left(\dfrac{3}{x-2}\right), & x \ne 2 \\ 4, & x = 2 \end{cases}[/latex];   [latex]g(0); \, g(-4); \, g(2)[/latex]

For the following exercises (21-22), determine whether the statement is true or false. Explain why.

  1. [latex]g(x)=\sqrt[3]{x}[/latex] is an odd root function
  2. A function of the form [latex]f(x)=x^b[/latex], where [latex]b[/latex] is a real valued constant, is an exponential function.

For the following exercises (23-27), complete all parts of each question. You may utilize calculators or any technological aids to facilitate your calculations.

  1. A company purchases some computer equipment for [latex]$20,500[/latex]. At the end of a [latex]3[/latex]-year period, the value of the equipment has decreased linearly to [latex]$12,300[/latex].
    1. Find a function [latex]y=V(t)[/latex] that determines the value [latex]V[/latex] of the equipment at the end of [latex]t[/latex] years.
    2. Find and interpret the meaning of the [latex]x[/latex]– and [latex]y[/latex]-intercepts for this situation.
    3. What is the value of the equipment at the end of [latex]5[/latex] years?
    4. When will the value of the equipment be [latex]$3000[/latex]?
  2. A family bakery makes cupcakes and sells them at local outdoor festivals. For a music festival, there is a fixed cost of [latex]$125[/latex] to set up a cupcake stand. The owner estimates that it costs [latex]$0.75[/latex] to make each cupcake. The owner is interested in determining the total cost [latex]C[/latex] as a function of number of cupcakes made.
    1. Find a linear function that relates cost [latex]C[/latex] to [latex]x[/latex], the number of cupcakes made.
    2. Find the cost to bake [latex]160[/latex] cupcakes.
    3. If the owner sells the cupcakes for [latex]$1.50[/latex] apiece, how many cupcakes does she need to sell to start making profit? (Hint: Use the INTERSECTION function on a calculator to find this number.)
  3. A car was purchased for [latex]$26,000[/latex]. The value of the car depreciates by [latex]$1500[/latex] per year.
    1. Find a linear function that models the value [latex]V[/latex] of the car after [latex]t[/latex] years.
    2. Find and interpret [latex]V(4)[/latex].
  4. The total cost [latex]C[/latex] (in thousands of dollars) to produce a certain item is modeled by the function [latex]C(x)=10.50x+28,500[/latex], where [latex]x[/latex] is the number of items produced. Determine the cost to produce [latex]175[/latex] items.
  5. The output (as a percent of total capacity) of nuclear power plants in the United States can be modeled by the function [latex]P(t)=1.8576t+68.052[/latex], where [latex]t[/latex] is time in years and [latex]t=0[/latex] corresponds to the beginning of 2000. Use the model to predict the percentage output in 2015.