Chapter 3 Functions
Chapter 3 Review Exercises
Functions and Function Notation
For the following exercises, determine whether the relation is a function.
- [latex]\left\{\left(a,b\right),\left(c,d\right),\left(e,d\right)\right\}[/latex]
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function
- [latex]\left\{\left(5,2\right),\left(6,1\right),\left(6,2\right),\left(4,8\right)\right\}[/latex]
- [latex]{y}^{2}+4=x,\,[/latex]for[latex]\,x\,[/latex] the independent variable and [latex]\,y\,[/latex] the dependent variable
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not a function
- Is the graph in Figure 1 a function?
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]
- [latex]f\left(x\right)=-2{x}^{2}+3x[/latex]
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[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]
- [latex]f\left(x\right)=2|3x-1|[/latex]
For the following exercises, determine whether the functions are one-to-one.
- [latex]f\left(x\right)=-3x+5[/latex]
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one-to-one
- [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.
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function
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function
For the following exercises, graph the functions.
- [latex]f\left(x\right)=|x+1|[/latex]
- [latex]f\left(x\right)={x}^{2}-2[/latex]
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For the following exercises, use Figure 2 to approximate the values.
- [latex]f\left(2\right)[/latex]
- [latex]f\left(-2\right)[/latex]
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[latex]2[/latex]
- If [latex]\,f\left(x\right)=-2,\,[/latex] then solve for [latex]\,x.[/latex]
- If [latex]\,f\left(x\right)=1,\,[/latex] then solve for [latex]\,x.[/latex]
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[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.
- [latex]\frac{h\left(2\right)-h\left(1\right)}{2-1}[/latex]
- [latex]\frac{h\left(a\right)-h\left(1\right)}{a-1}[/latex]
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[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.
- [latex]f\left(x\right)=\frac{2}{3x+2}[/latex]
- [latex]f\left(x\right)=\frac{x-3}{{x}^{2}-4x-12}[/latex]
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[latex]\left(-\infty ,-2\right)\cup \left(-2,6\right)\cup \left(6,\infty \right)[/latex]
- [latex]f\left(x\right)=\frac{\sqrt{x-6}}{\sqrt{x-4}}[/latex]
- 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]
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Rates of Change and Behavior of Graphs
For the following exercises, find the average rate of change of the functions from[latex]\