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Studienführer > MTH 163, Precalculus

Section Exercises

1. Describe why the horizontal line test is an effective way to determine whether a function is one-to-one? 2. Why do we restrict the domain of the function [latex]f\left(x\right)={x}^{2}[/latex] to find the function’s inverse? 3. Can a function be its own inverse? Explain. 4. Are one-to-one functions either always increasing or always decreasing? Why or why not? 5. How do you find the inverse of a function algebraically? 6. Show that the function [latex]f\left(x\right)=a-x[/latex] is its own inverse for all real numbers [latex]a[/latex]. For the following exercises, find [latex]{f}^{-1}\left(x\right)[/latex] for each function. 7. [latex]f\left(x\right)=x+3[/latex] 8. [latex]f\left(x\right)=x+5[/latex] 9. [latex]f\left(x\right)=2-x[/latex] 10. [latex]f\left(x\right)=3-x[/latex] 11. [latex]f\left(x\right)=\frac{x}{x+2}\\[/latex] 12. [latex]f\left(x\right)=\frac{2x+3}{5x+4}[/latex] For the following exercises, find a domain on which each function [latex]f[/latex] is one-to-one and non-decreasing. Write the domain in interval notation. Then find the inverse of [latex]f[/latex] restricted to that domain. 13. [latex]f\left(x\right)={\left(x+7\right)}^{2}[/latex] 14. [latex]f\left(x\right)={\left(x - 6\right)}^{2}[/latex] 15. [latex]f\left(x\right)={x}^{2}-5[/latex] 16. Given [latex]f\left(x\right)=\frac{x}{2}+x\\[/latex] and [latex]g\left(x\right)=\frac{2x}{1-x}\\[/latex] a. Find [latex]f\left(g\left(x\right)\right)[/latex] and [latex]g\left(f\left(x\right)\right)\\[/latex] b. What does the answer tell us about the relationship between [latex]f\left(x\right)[/latex] and [latex]g\left(x\right)?[/latex] For the following exercises, use function composition to verify that [latex]f\left(x\right)[/latex] and [latex]g\left(x\right)[/latex] are inverse functions. 17. [latex]f\left(x\right)=\sqrt[3]{x - 1}[/latex] and [latex]g\left(x\right)={x}^{3}+1[/latex] 18. [latex]f\left(x\right)=-3x+5[/latex] and [latex]g\left(x\right)=\frac{x - 5}{-3}[/latex] For the following exercises, use a graphing utility to determine whether each function is one-to-one. 19. [latex]f\left(x\right)=\sqrt{x}[/latex] 20. [latex]f\left(x\right)=\sqrt[3]{3x+1}[/latex] 21. [latex]f\left(x\right)=-5x+1[/latex] 22. [latex]f\left(x\right)={x}^{3}-27[/latex] For the following exercises, determine whether the graph represents a one-to-one function. 23. Graph of an upright parabola with vertex at (10, -10), passing through (0,0) and (25,0) 24. Graph of a step-function, with y = -5 for {x|-10<=x,0} and y = 0 for {x|0<=x<10} For the following exercises, use the graph of [latex]f[/latex] shown in [link]. Graph of the line y = (-3/2)x + 3 25. Find [latex]f\left(0\right)[/latex]. 26. Solve [latex]f\left(x\right)=0[/latex]. 27. Find [latex]{f}^{-1}\left(0\right)[/latex]. 28. Solve [latex]{f}^{-1}\left(x\right)=0[/latex]. For the following exercises, use the graph of the one-to-one function shown below. Graph of a square root function for {x|x>=2} 29. Sketch the graph of [latex]{f}^{-1}[/latex]. 30. Find [latex]f\left(6\right)\text{ and }{f}^{-1}\left(2\right)[/latex]. 31. If the complete graph of [latex]f[/latex] is shown, find the domain of [latex]f[/latex]. 32. If the complete graph of [latex]f[/latex] is shown, find the range of [latex]f[/latex]. For the following exercises, evaluate or solve, assuming that the function [latex]f[/latex] is one-to-one. 33. If [latex]f\left(6\right)=7[/latex], find [latex]{f}^{-1}\left(7\right)[/latex]. 34. If [latex]f\left(3\right)=2[/latex], find [latex]{f}^{-1}\left(2\right)[/latex]. 35. If [latex]{f}^{-1}\left(-4\right)=-8[/latex], find [latex]f\left(-8\right)[/latex]. 36. If [latex]{f}^{-1}\left(-2\right)=-1[/latex], find [latex]f\left(-1\right)[/latex]. For the following exercises, use the values listed in the table below to evaluate or solve.
[latex]x[/latex] [latex]f\left(x\right)[/latex]
0 8
1 0
2 7
3 4
4 2
5 6
6 5
7 3
8 9
9 1
37. Find [latex]f\left(1\right)[/latex]. 38. Solve [latex]f\left(x\right)=3[/latex]. 39. Find [latex]{f}^{-1}\left(0\right)[/latex]. 40. Solve [latex]{f}^{-1}\left(x\right)=7[/latex]. 41. Use the tabular representation of [latex]f[/latex] to create a table for [latex]{f}^{-1}\left(x\right)[/latex].
[latex]x[/latex] 3 6 9 13 14
[latex]f\left(x\right)[/latex] 1 4 7 12 16
For the following exercises, find the inverse function. Then, graph the function and its inverse. 42. [latex]f\left(x\right)=\frac{3}{x - 2}[/latex] 43. [latex]f\left(x\right)={x}^{3}-1[/latex] 44. Find the inverse function of [latex]f\left(x\right)=\frac{1}{x - 1}[/latex]. Use a graphing utility to find its domain and range. Write the domain and range in interval notation. 45. To convert from [latex]x[/latex] degrees Celsius to [latex]y[/latex] degrees Fahrenheit, we use the formula [latex]f\left(x\right)=\frac{9}{5}x+32[/latex]. Find the inverse function, if it exists, and explain its meaning. 46. The circumference [latex]C[/latex] of a circle is a function of its radius given by [latex]C\left(r\right)=2\pi r[/latex]. Express the radius of a circle as a function of its circumference. Call this function [latex]r\left(C\right)[/latex]. Find [latex]r\left(36\pi \right)[/latex] and interpret its meaning. 47. A car travels at a constant speed of 50 miles per hour. The distance the car travels in miles is a function of time, [latex]t[/latex], in hours given by [latex]d\left(t\right)=50t[/latex]. Find the inverse function by expressing the time of travel in terms of the distance traveled. Call this function [latex]t\left(d\right)[/latex]. Find [latex]t\left(180\right)[/latex] and interpret its meaning.

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  • Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..