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Chapter 2: Problem 73

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

Short Answer

Expert verified
To graph the inverse of a function, the graph of the original function is reflected over the line \(y = x\). For every point on the original graph \((a, b)\), a corresponding point on the inverse graph is situated at \((b, a)\).

Step by step solution

01

Understand the Concept of Inverse Functions

An inverse function is essentially the 'opposite' of the original function. If the original function gives you an output 'y' for an input 'x', the inverse function will take the 'y' and give you back the original 'x'. Algebraically, if a function \(f\) takes \(x\) to \(y\) (i.e., \(f(x) = y\)), its inverse denoted by \(f^{-1}\) takes \(y\) back to \(x\) (i.e., \(f^{-1}(y) = x\)). On a graph, this corresponds to flipping over the line \(y = x\).
02

Plot the Original Function

Start by graphing the original function. This could be any one-to-one function, meaning every 'x' value corresponds to one unique 'y' value and vice versa. Note the key points on this graph.
03

Draw the line \(y = x\)

This line acts as the 'mirror' about which the function and its inverse are reflected. It will help you successfully plot the inverse function.
04

Reflect the Original Function

To graph the inverse function, reflect each point of the original function over the line \(y = x\). That is, if there鈥檚 a point \((a, b)\) on the original function, then create a point \((b, a)\) for the inverse function.
05

Connect the Points

Draw the curve of the inverse function by connecting the reflected points. This curve should be a mirror image of the original function curve across the \(y=x\) line.

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Most popular questions from this chapter

Begin by graphing the absolute value function, \(f(x)-|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)--2|x+4|+1 $$

Explain how to determine if two functions are inverses of each other.

How can a graphing utility be used to visually determine if two functions are inverses of each other?

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. $$ f(x)=4 x+4, g(x)=0.25 x-1 $$

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. $$ f(x)=\sqrt[3]{x}-2, g(x)=(x+2)^{3} $$
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