New Zealand
Level 8 - NCEA Level 3

# Graphs of Inverse Functions

Lesson

When we think about the graphs of inverse functions, geometrically we are talking about reflecting the function $f(x)$f(x) over the line $y=x$y=x to draw the inverse function $f^{-1}\left(x\right)$f1(x).

By doing this the $x$x values, or the inputs, become the $y$y values, or the outputs, and vice versa.

Let's start by reflecting a few points, belonging to a curve, over the line $y=x$y=x to see this in action.

We can see that $(0,2)$(0,2) is reflected across to $(2,0)$(2,0)

Similarly $(1,3)$(1,3) is transformed to $(3,1)$(3,1) and $(2,6)$(2,6) to $(6,2)$(6,2).

So when we want to use the graph of $f(x)$f(x) to draw its inverse $f^{-1}\left(x\right)$f1(x), then we want to take points on the curve of $f(x)$f(x) and reflect them over the line $y=x$y=x.

Let's take a look at an example.

##### example

The function $f\left(x\right)=\sqrt{x+3}+1$f(x)=x+3+1 is graphed below along with the line $y=x$y=x. Sketch the graph of $f^{-1}\left(x\right)$f1(x).

Think: We'll first identify some points on $f(x)$f(x) and then reflect them over $y=x$y=x.

Do:

#### Worked Examples

##### example 1

Below we have sketched the line $y=\frac{1}{2}x$y=12x (labelled $B$B) over the line $y=x$y=x (labelled $A$A).

1. By reflecting $y=\frac{1}{2}x$y=12x about the line $y=x$y=x, graph the inverse of $y=\frac{1}{2}x$y=12x.

##### example 2

Below we have sketched the line $y=\frac{x^2}{4}+1$y=x24+1 as defined for $x\le0$x0 (labelled $B$B) over the line $y=x$y=x (labelled $A$A).

1. By reflecting this arm of $y=\frac{x^2}{4}+1$y=x24+1 about the line $y=x$y=x, graph the inverse of the arm of $y=\frac{x^2}{4}+1$y=x24+1 defined over $x\le0$x0.

##### example 3

Below we have graphed the line $y=\left(\frac{3}{2}\right)^{-x}$y=(32)x (labelled $B$B) over the line $y=x$y=x (labelled $A$A).

1. By reflecting this arm of $y=\left(\frac{3}{2}\right)^{-x}$y=(32)x about the line $y=x$y=x, graph the inverse of the arm of $y=\left(\frac{3}{2}\right)^{-x}$y=(32)x.