How To Draw The Inverse Of A Graph

At present that nosotros tin find the inverse of a office, we will explore the graphs of functions and their inverses. Let us return to the quadratic function [latex]f\left(ten\right)={x}^{2}[/latex] restricted to the domain [latex]\left[0,\infty \correct)[/latex], on which this part is i-to-one, and graph it as in Figure 7.

Graph of f(x).

Figure 7. Quadratic office with domain restricted to [0, ∞).

Restricting the domain to [latex]\left[0,\infty \right)[/latex] makes the office ane-to-one (it will obviously pass the horizontal line test), so it has an inverse on this restricted domain.

We already know that the inverse of the toolkit quadratic office is the foursquare root function, that is, [latex]{f}^{-1}\left(x\right)=\sqrt{x}[/latex]. What happens if nosotros graph both [latex]f\text{ }[/latex] and [latex]{f}^{-1}[/latex] on the same set up of axes, using the [latex]x\text{-}[/latex] axis for the input to both [latex]f\text{ and }{f}^{-1}?[/latex]

Nosotros find a distinct relationship: The graph of [latex]{f}^{-1}\left(10\right)[/latex] is the graph of [latex]f\left(10\right)[/latex] reflected nearly the diagonal line [latex]y=x[/latex], which we volition call the identity line, shown in Figure 8.

Graph of f(x) and f^(-1)(x).

Figure 8. Foursquare and square-root functions on the non-negative domain

This human relationship will be observed for all one-to-one functions, because information technology is a result of the function and its inverse swapping inputs and outputs. This is equivalent to interchanging the roles of the vertical and horizontal axes.

Example 10: Finding the Inverse of a Function Using Reflection about the Identity Line

Given the graph of [latex]f\left(x\correct)[/latex], sketch a graph of [latex]{f}^{-1}\left(x\right)[/latex].

Graph of f^(-1)(x).

Figure ix

Solution

This is a 1-to-one function, and then nosotros will be able to sketch an inverse. Note that the graph shown has an apparent domain of [latex]\left(0,\infty \correct)[/latex] and range of [latex]\left(-\infty ,\infty \right)[/latex], and then the changed volition have a domain of [latex]\left(-\infty ,\infty \correct)[/latex] and range of [latex]\left(0,\infty \right)[/latex].

If we reflect this graph over the line [latex]y=x[/latex], the point [latex]\left(ane,0\correct)[/latex] reflects to [latex]\left(0,1\right)[/latex] and the signal [latex]\left(4,two\right)[/latex] reflects to [latex]\left(2,4\right)[/latex]. Sketching the inverse on the same axes as the original graph gives us the upshot in Figure 10.

Graph of f(x) and f^(-1)(x).

Figure x. The function and its inverse, showing reflection near the identity line

Try Information technology nine

Draw graphs of the functions [latex]f\text{ }[/latex] and [latex]\text{ }{f}^{-1}[/latex].

Solution

Q & A

Is there whatsoever function that is equal to its own inverse?

Yep. If [latex]f={f}^{-one}[/latex], and then [latex]f\left(f\left(10\right)\right)=ten[/latex], and we tin think of several functions that have this belongings. The identity function does, then does the reciprocal part, considering

[latex]\frac{1}{\frac{one}{x}}=x[/latex]

Any office [latex]f\left(ten\right)=c-ten[/latex], where [latex]c[/latex] is a constant, is also equal to its ain changed.