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 $f^{-1}\left(x\right)$`f`−1(`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 $f^{-1}\left(x\right)$`f`−1(`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.

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 $f^{-1}\left(x\right)$`f`−1(`x`).

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

Do:

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

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By reflecting $y=\frac{1}{2}x$

`y`=12`x`about the line $y=x$`y`=`x`, graph the inverse of $y=\frac{1}{2}x$`y`=12`x`.Loading Graph...

Below we have sketched the line $y=\frac{x^2}{4}+1$`y`=`x`24+1 as defined for $x\le0$`x`≤0 (labelled $B$`B`) over the line $y=x$`y`=`x` (labelled $A$`A`).

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By reflecting this arm of $y=\frac{x^2}{4}+1$

`y`=`x`24+1 about the line $y=x$`y`=`x`, graph the inverse of the arm of $y=\frac{x^2}{4}+1$`y`=`x`24+1 defined over $x\le0$`x`≤0.Loading Graph...

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`).

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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`.Loading Graph...

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions