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Chapter 8: Problem 72

Describe how to use the graph of a one-to-one function to draw the graph of its inverse function.

Short Answer

Expert verified
The graph of an inverse function can be drawn by reflecting the graph of the original function across the line \(y = x\). Each point \( (a, b) \) on the original graph corresponds to a point \( (b, a) \) on the graph of the inverse function.

Step by step solution

01

Note the function graph

Study the graph of the original function. The graph should be a one-to-one function, meaning every x-value should correspond to exactly one y-value in order for it to have an inverse.
02

Draw the line y = x

Draw the line \(y = x\) on the same graph. This line will be the mirror line that the original function is reflected across to get the inverse function.
03

Reflect the original function

Reflect each point of the original function's graph across the line \(y = x\) to form the graph of its inverse function. Essentially, if a point \((a, b)\) is on the graph of a function, then the point \((b, a)\) is on the graph of its inverse.
04

Sketch the graph of the inverse

Sketch the new graph that represents the inverse function based on the points obtained from the reflection.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I'm working with the linear function \(f(x)=3 x+5\) and I do not need to find \(f^{-1}\) in order to determine the value of \(\left(f \circ f^{-1}\right)(17)\).

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(f(x)=3 x,\) then \(f^{-1}(x)=\frac{1}{3 x}\).

Use a graphing utility to graph \(f\) and \(g\) in the same viewing rectangle. In addition, graph the line \(y=x\) and visually determine if \(f\) and g are inverses. $$f(x)=\frac{1}{x}+2, \quad g(x)=\frac{1}{x-2}$$

Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=\frac{2}{x-5} \quad \text { and } \quad g(x)=\frac{2}{x}+5$$

Will help you prepare for the material covered in the first section of the next chapter. Solve and express the solution set in interval notation: $$600 x-(500,000+400 x)>0$$
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