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Chapter 3 Functions

Chapter 3 Review Exercises

Functions and Function Notation

For the following exercises, determine whether the relation is a function.

  1. [latex]\left\{\left(a,b\right),\left(c,d\right),\left(e,d\right)\right\}[/latex]
Show Solution

function

  1. [latex]\left\{\left(5,2\right),\left(6,1\right),\left(6,2\right),\left(4,8\right)\right\}[/latex]
  1. [latex]{y}^{2}+4=x,\,[/latex]for[latex]\,x\,[/latex] the independent variable and [latex]\,y\,[/latex] the dependent variable
Show Solution

not a function

  1. Is the graph in Figure 1 a function?
Graph of a parabola
Figure 1

For the following exercises, evaluate the function at the indicated values:[latex]\,\,\,f\left(-3\right);\,\,f\left(2\right);\,\,\,f\left(-a\right);\,\,\,-f\left(a\right);\,\,\,f\left(a+h\right).[/latex]

  1. [latex]f\left(x\right)=-2{x}^{2}+3x[/latex]
Show Solution

[latex]f\left(-3\right)=-27;[/latex][latex]f\left(2\right)=-2;[/latex][latex]f\left(-a\right)=-2{a}^{2}-3a;[/latex]

[latex]-f\left(a\right)=2{a}^{2}-3a;[/latex][latex]f\left(a+h\right)=-2{a}^{2}+3a-4ah+3h-2{h}^{2}[/latex]

  1. [latex]f\left(x\right)=2|3x-1|[/latex]

For the following exercises, determine whether the functions are one-to-one.

  1. [latex]f\left(x\right)=-3x+5[/latex]
Show Solution

one-to-one

  1. [latex]f\left(x\right)=|x-3|[/latex]

For the following exercises, use the vertical line test to determine if the relation whose graph is provided is a function.

Graph of a cubic function
Show Solution

function

Graph of a relation.
Graph of a relation
Show Solution

function

For the following exercises, graph the functions.

  1. [latex]f\left(x\right)=|x+1|[/latex]
  1. [latex]f\left(x\right)={x}^{2}-2[/latex]
Show Solution

Graph of f(x)

 

For the following exercises, use Figure 2 to approximate the values.

Graph of f(x)
Figure 2
  1. [latex]f\left(2\right)[/latex]
  1. [latex]f\left(-2\right)[/latex]
Show Solution

[latex]2[/latex]

  1. If [latex]\,f\left(x\right)=-2,\,[/latex] then solve for [latex]\,x.[/latex]
  1. If [latex]\,f\left(x\right)=1,\,[/latex] then solve for [latex]\,x.[/latex]
Show Solution

[latex]x=-1.8\text{ }[/latex]or[latex]\text{ }x=1.8[/latex]

For the following exercises, use the function [latex]\,h\left(t\right)=-16{t}^{2}+80t\,[/latex] to find the values in simplest form.

  1. [latex]\frac{h\left(2\right)-h\left(1\right)}{2-1}[/latex]
  1. [latex]\frac{h\left(a\right)-h\left(1\right)}{a-1}[/latex]
Show Solution

[latex]\frac{-64+80a-16{a}^{2}}{-1+a}=-16a+64[/latex]

Domain and Range

For the following exercises, find the domain of each function, expressing answers using interval notation.

  1. [latex]f\left(x\right)=\frac{2}{3x+2}[/latex]
  1. [latex]f\left(x\right)=\frac{x-3}{{x}^{2}-4x-12}[/latex]
Show Solution

[latex]\left(-\infty ,-2\right)\cup \left(-2,6\right)\cup \left(6,\infty \right)[/latex]

  1. [latex]f\left(x\right)=\frac{\sqrt{x-6}}{\sqrt{x-4}}[/latex]
  1. Graph this piecewise function:[latex]f\left(x\right)=\bigg\{\begin{array}{l}x+1\text{ } \ \ \ \ \ \ \ \ x<-2\\ -2x-3\text{ } \ \ \ x\ge -2\end{array}[/latex]
Show Solution

Graph of f(x)

 

Rates of Change and Behavior of Graphs

For the following exercises, find the average rate of change of the functions from[latex]\,x=1\text{ to }x=2.[/latex]

  1. [latex]f\left(x\right)=4x-3[/latex]
  1. [latex]f\left(x\right)=10{x}^{2}+x[/latex]
Show Solution

[latex]31[/latex]

  1. [latex]f\left(x\right)=-\frac{2}{{x}^{2}}[/latex]

For the following exercises, use the graphs to determine the intervals on which the functions are increasing, decreasing, or constant.

Graph of a parabola
Show Solution

increasing[latex]\,\left(2,\infty \right);\,[/latex]
decreasing[latex]\,\left(-\infty ,2\right)[/latex]

Graph of a cubic function.
Graph of a function.
Show Solution

increasing[latex]\text{}\left(-3,1\right); \text{}[/latex]constant[latex]\,\left(-\infty ,-3\right)\cup \left(1,\infty \right)[/latex]

  1. Find the local minimum of the function graphed in  Exercise 27.
  1. Find the local extrema for the function graphed in Exercise 28.
Show Solution

local minimum[latex]\,\left(-2,-3\right);\,[/latex]local maximum[latex]\,\left(1,3\right)[/latex]

  1. For the graph in Figure 3 below, the domain of the function is [latex]\,\left[-3,3\right].[/latex]The range is [latex]\,\left[-10,10\right].\,[/latex] Find the absolute minimum of the function on this interval.
  1. Find the absolute maximum of the function graphed in Figure 3.
Graph of a cubic function.
Figure 3
Show Solution

[latex]\,\left(-1.8,10\right)\,[/latex]

Composition of Functions

For the following exercises, find [latex]\,\left(f\circ g\right)\left(x\right)\,[/latex] and [latex]\,\left(g\circ f\right)\left(x\right)\,[/latex] for each pair of functions.

  1. [latex]f\left(x\right)=4-x,\,g\left(x\right)=-4x[/latex]
  1. [latex]f\left(x\right)=3x+2,\,g\left(x\right)=5-6x[/latex]
Show Solution

[latex]\left(f\circ g\right)\left(x\right)=17-18x;\,\left(g\circ f\right)\left(x\right)=-7-18x[/latex]

  1. [latex]f\left(x\right)={x}^{2}+2x,\,g\left(x\right)=5x+1[/latex]
  1. [latex]f\left(x\right)=\sqrt{x+2},\text{ }g\left(x\right)=\frac{1}{x}[/latex]
Show Solution

[latex]\left(f\circ g\right)\left(x\right)=\sqrt{\frac{1}{x}+2};\,\left(g\circ f\right)\left(x\right)=\frac{1}{\sqrt{x+2}}[/latex]

  1. [latex]\,f\left(x\right)=\frac{x+3}{2},\text{ }g\left(x\right)=\sqrt{1-x}\,[/latex]

For the following exercises, find [latex]\,\left(f\circ g\right)\,[/latex] and the domain for [latex]\,\left(f\circ g\right)\left(x\right)\,[/latex]for each pair of functions.

  1. [latex]f\left(x\right)=\frac{x+1}{x+4},\text{ }g\left(x\right)=\frac{1}{x}[/latex]
Show Solution

[latex]\left(f\circ g\right)\left(x\right)=\frac{1+x}{1+4x}, x\ne 0, x\ne -\frac{1}{4}[/latex]

  1. [latex]f\left(x\right)=\frac{1}{x+3},\text{ }g\left(x\right)=\frac{1}{x-9}[/latex]
  1. [latex]f\left(x\right)=\frac{1}{x},\text{ }g\left(x\right)=\sqrt{x}[/latex]
Show Solution

[latex]\left(f\circ g\right)\left(x\right)=\frac{1}{\sqrt{x}},\,x>0[/latex]

  1. [latex]f\left(x\right)=\frac{1}{{x}^{2}-1},\text{ }g\left(x\right)=\sqrt{x+1}[/latex]

For the following exercises, express each function[latex]\,H\,[/latex] as a composition of two functions [latex]\,f\,[/latex]and[latex]\,g\,[/latex] where [latex]\,H\left(x\right)=\left(f\circ g\right)\left(x\right).[/latex]

  1. [latex]H\left(x\right)=\sqrt{\frac{2x-1}{3x+4}}[/latex]
Show Solution

sample:[latex]\,g\left(x\right)=\frac{2x-1}{3x+4};\,f\left(x\right)=\sqrt{x}[/latex]

  1. [latex]H\left(x\right)=\frac{1}{{\left(3{x}^{2}-4\right)}^{-3}}[/latex]

Transformation of Functions

For the following exercises, sketch a graph of the given function.

  1. [latex]f\left(x\right)={\left(x-3\right)}^{2}[/latex]
Show Solution

Graph of f(x)

 

  1. [latex]f\left(x\right)={\left(x+4\right)}^{3}[/latex]
  1. [latex]f\left(x\right)=\sqrt{x}+5[/latex]
Show Solution

Graph of f(x)

 

  1. [latex]f\left(x\right)=-{x}^{3}[/latex]
  1. [latex]f\left(x\right)=\sqrt[3]{-x}[/latex]
Show Solution

Graph of f(x)

 

  1. [latex]f\left(x\right)=5\sqrt{-x}-4[/latex]
  1. [latex]f\left(x\right)=4\left[|x-2|-6\right][/latex]
Show Solution

Graph of f(x)

 

  1. [latex]f\left(x\right)=-{\left(x+2\right)}^{2}-1[/latex]

For the following exercises, sketch the graph of the function [latex]\,g\,[/latex] if the graph of the function [latex]\,f\,[/latex] is shown in Figure 4.

Graph of f(x)
Figure 4
  1. [latex]g\left(x\right)=f\left(x-1\right)[/latex]
Show Solution

Graph of a half circle.

 

  1. [latex]g\left(x\right)=3f\left(x\right)[/latex]

For the following exercises, write the equation for the standard function represented by each of the graphs below.

Graph of an absolute function
Show Solution

[latex]f\left(x\right)=|x-3|[/latex]

Graph of a half circle

For the following exercises, determine whether each function below is even, odd, or neither.

  1. [latex]f\left(x\right)=3{x}^{4}[/latex]
Show Solution

even

  1. [latex]g\left(x\right)=\sqrt{x}[/latex]
  1. [latex]h\left(x\right)=\frac{1}{x}+3x[/latex]
Show Solution

odd

For the following exercises, analyze the graph and determine whether the graphed function is even, odd, or neither.

Graph of a parabola.
Graph of a parabola
Show Solution

even

Graph of a cubic function

Absolute Value Functions

For the following exercises, write an equation for the transformation of[latex]\,f\left(x\right)=|x|.[/latex]

Graph of f(x).
Show Solution

[latex]f\left(x\right)=\frac{1}{2}|x+2|+1[/latex]

Graph of f(x)
Graph of f(x).
Show Solution

[latex]f\left(x\right)=-3|x-3|+3[/latex]

For the following exercises, graph the absolute value function.

  1. [latex]f\left(x\right)=|x-5|[/latex]
  1. [latex]f\left(x\right)=-|x-3|[/latex]
Show Solution

Graph of f(x)

 

  1. [latex]f\left(x\right)=|2x-4|[/latex]

Inverse Functions

For the following exercises, find[latex]\text{ }{f}^{-1}\left(x\right)\text{ }[/latex]for each function.

  1. [latex]f\left(x\right)=9+10x[/latex]
  1. [latex]f\left(x\right)=\frac{x}{x+2}[/latex]
Show Solution

[latex]{f}^{-1}\left(x\right)=\frac{-2x}{x-1}[/latex]

For the following exercise, find a domain on which the function[latex]\text{ }f\text{ }[/latex]is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of[latex]\text{ }f\text{ }[/latex]restricted to that domain.

  1. [latex]f\left(x\right)={x}^{2}+1[/latex]
  1. Given [latex]f\left(x\right)={x}^{3}-5[/latex] and [latex]g\left(x\right)=\sqrt[3]{x+5}:[/latex]
  1. Find [latex]f\left(g\left(x\right)\right)[/latex] and [latex]g\left(f\left(x\right)\right).[/latex]
  2. What does the answer tell us about the relationship between [latex]f\left(x\right)[/latex] and [latex]g\left(x\right)?[/latex]
Show Solution
  1. [latex]f\left(g\left(x\right)\right)=x[/latex] and [latex]g\left(f\left(x\right)\right)=x.[/latex]
  2. This tells us that [latex]f[/latex] and [latex]g[/latex] are inverse functions

For the following exercises, use a graphing utility to determine whether each function is one-to-one.

  1. [latex]f\left(x\right)=\frac{1}{x}[/latex]
Show Solution

The function is one-to-one.

Graph ofa function

 

 

  1. [latex]f\left(x\right)=-3{x}^{2}+x[/latex]
Show Solution

The function is not one-to-one.

Graph of a function

 

  1. If [latex]f\left(5\right)=2,[/latex] find [latex]{f}^{-1}\left(2\right).[/latex]
Show Solution
[latex]5[/latex]

  1. If [latex]f\left(1\right)=4,[/latex] find [latex]{f}^{-1}\left(4\right).[/latex]

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