Section Exercises
1. What is the difference between a relation and a function? 2. What is the difference between the input and the output of a function? 3. Why does the vertical line test tell us whether the graph of a relation represents a function? 4. How can you determine if a relation is a one-to-one function? 5. Why does the horizontal line test tell us whether the graph of a function is one-to-one? For the following exercises, determine whether the relation represents a function. 6. [latex]\left\{\left(a,b\right),\text{ }\left(c,d\right),\text{ }\left(a,c\right)\right\}[/latex] 7. [latex]\left\{\left(a,b\right),\left(b,c\right),\left(c,c\right)\right\}[/latex] For the following exercises, determine whether the relation represents [latex]y[/latex] as a function of [latex]x[/latex]. 8. [latex]5x+2y=10[/latex] 9. [latex]y={x}^{2}[/latex] 10. [latex]x={y}^{2}[/latex] 11. [latex]3{x}^{2}+y=14[/latex] 12. [latex]2x+{y}^{2}=6[/latex] 13. [latex]y=-2{x}^{2}+40x[/latex] 14. [latex]y=\frac{1}{x}[/latex] 15. [latex]x=\frac{3y+5}{7y - 1}[/latex] 16. [latex]x=\sqrt{1-{y}^{2}}[/latex] 17. [latex]y=\frac{3x+5}{7x - 1}[/latex] 18. [latex]{x}^{2}+{y}^{2}=9[/latex] 19. [latex]2xy=1[/latex] 20. [latex]x={y}^{3}[/latex] 21. [latex]y={x}^{3}[/latex] 22. [latex]y=\sqrt{1-{x}^{2}}[/latex] 23. [latex]x=\pm \sqrt{1-y}[/latex] 24. [latex]y=\pm \sqrt{1-x}[/latex] 25. [latex]{y}^{2}={x}^{2}[/latex] 26. [latex]{y}^{3}={x}^{2}[/latex] For the following exercises, evaluate the function [latex]f[/latex] at the indicated values [latex]\text{ }f\left(-3\right),f\left(2\right),f\left(-a\right),-f\left(a\right),f\left(a+h\right)[/latex]. 27. [latex]f\left(x\right)=2x - 5[/latex] 28. [latex]f\left(x\right)=-5{x}^{2}+2x - 1[/latex] 29. [latex]f\left(x\right)=\sqrt{2-x}+5[/latex] 30. [latex]f\left(x\right)=\frac{6x - 1}{5x+2}[/latex] 31. [latex]f\left(x\right)=|x - 1|-|x+1|[/latex] 32. Given the function [latex]g\left(x\right)=5-{x}^{2}[/latex], evaluate [latex]\frac{g\left(x+h\right)-g\left(x\right)}{h},h\ne 0[/latex]. 33. Given the function [latex]g\left(x\right)={x}^{2}+2x[/latex], evaluate [latex]\frac{g\left(x\right)-g\left(a\right)}{x-a},x\ne a[/latex]. 34. Given the function [latex]k\left(t\right)=2t - 1:[/latex]
a. Evaluate [latex]k\left(2\right)[/latex]. b. Solve [latex]k\left(t\right)=7[/latex].
35. Given the function [latex]f\left(x\right)=8 - 3x:[/latex]a. Evaluate [latex]f\left(-2\right)[/latex]. b. Solve [latex]f\left(x\right)=-1[/latex].
36. Given the function [latex]p\left(c\right)={c}^{2}+c:[/latex]a. Evaluate [latex]p\left(-3\right)[/latex]. b. Solve [latex]p\left(c\right)=2[/latex].
37. Given the function [latex]f\left(x\right)={x}^{2}-3x:[/latex]a. Evaluate [latex]f\left(5\right)[/latex]. b. Solve [latex]f\left(x\right)=4[/latex].
38. Given the function [latex]f\left(x\right)=\sqrt{x+2}:[/latex]a. Evaluate [latex]f\left(7\right)[/latex]. b. Solve [latex]f\left(x\right)=4[/latex].
39. Consider the relationship [latex]3r+2t=18[/latex].a. Write the relationship as a function [latex]r=f\left(t\right)[/latex]. b. Evaluate [latex]f\left(-3\right)[/latex]. c. Solve [latex]f\left(t\right)=2[/latex].
For the following exercises, use the vertical line test to determine which graphs show relations that are functions. 40.![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200527/CNX_Precalc_Figure_01_01_201.jpg)
![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200529/CNX_Precalc_Figure_01_01_202.jpg)
![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200530/CNX_Precalc_Figure_01_01_203.jpg)
![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200532/CNX_Precalc_Figure_01_01_204.jpg)
![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200533/CNX_Precalc_Figure_01_01_205.jpg)
![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200535/CNX_Precalc_Figure_01_01_206.jpg)
![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200537/CNX_Precalc_Figure_01_01_207.jpg)
![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200538/CNX_Precalc_Figure_01_01_208.jpg)
![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200539/CNX_Precalc_Figure_01_01_209.jpg)
![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200541/CNX_Precalc_Figure_01_01_210.jpg)
![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200543/CNX_Precalc_Figure_01_01_211.jpg)
![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200544/CNX_Precalc_Figure_01_01_212.jpg)
a. Evaluate [latex]f\left(-1\right)[/latex]. b. Solve for [latex]f\left(x\right)=3[/latex].
![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200546/CNX_Precalc_Figure_01_01_213.jpg)
a. Evaluate [latex]f\left(0\right)[/latex]. b. Solve for [latex]f\left(x\right)=-3[/latex].
![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200547/CNX_Precalc_Figure_01_01_214.jpg)
a. Evaluate [latex]f\left(4\right)[/latex]. b. Solve for [latex]f\left(x\right)=1[/latex].
![Graph of relation.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200549/CNX_Precalc_Figure_01_01_215.jpg)
![Graph of a circle.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200550/CNX_Precalc_Figure_01_01_216.jpg)
![Graph of a parabola.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200551/CNX_Precalc_Figure_01_01_232.jpg)
![Graph of a rotated cubic function.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200553/CNX_Precalc_Figure_01_01_217.jpg)
![Graph of half of 1/x.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200554/CNX_Precalc_Figure_01_01_218.jpg)
![Graph of a one-to-one function.](https://s3-us-west-2.amazonaws.com/courses-images-archive-read-only/wp-content/uploads/sites/924/2015/11/25200555/CNX_Precalc_Figure_01_01_233.jpg)
[latex]x[/latex] | 5 | 10 | 15 |
[latex]y[/latex] | 3 | 8 | 14 |
[latex]x[/latex] | 5 | 10 | 15 |
[latex]y[/latex] | 3 | 8 | 8 |
[latex]x[/latex] | 5 | 10 | 10 |
[latex]y[/latex] | 3 | 8 | 14 |
[latex]x[/latex] | [latex]f\left(x\right)[/latex] |
0 | 74 |
1 | 28 |
2 | 1 |
3 | 53 |
4 | 56 |
5 | 3 |
6 | 36 |
7 | 45 |
8 | 14 |
9 | 47 |
- [latex]f\left(x\right)=3x - 2[/latex]
- [latex]g\left(x\right)=5-{x}^{2}[/latex]
- [latex]h\left(x\right)=-2{x}^{2}+3x - 1[/latex]
a. The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function [latex]f[/latex]. b. Explain the meaning of the statement [latex]f\left(5\right)=2[/latex].
89. 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\left(a\right)[/latex].a. A garden with area 5000 ft2 requires 50 yd3 of dirt. Express this information in terms of the function [latex]g[/latex]. b. Explain the meaning of the statement [latex]g\left(100\right)=1[/latex].
90. Let [latex]f\left(t\right)[/latex] be the number of ducks in a lake [latex]t[/latex] years after 1990. Explain the meaning of each statement:a. [latex]f\left(5\right)=30[/latex] b. [latex]f\left(10\right)=40[/latex]
91. Let [latex]h\left(t\right)[/latex] be the height above ground, in feet, of a rocket [latex]t[/latex] seconds after launching. Explain the meaning of each statement:a. [latex]h\left(1\right)=200[/latex] b. [latex]h\left(2\right)=350[/latex]
92. Show that the function [latex]f\left(x\right)=3{\left(x - 5\right)}^{2}+7[/latex] is not one-to-one.