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1.10 Inverse relations and functions


Finding the inverse of a function

We have seen the word "inverse" before when we were solving equations. Recall that all of our basic operations have an inverse.


Addition and subtraction are inverse operations of one another

Multiplication and division are inverse operations of one another

Squaring and square rooting are inverse operations of one another

We now want to find the inverse of not just one operation, but an entire function.


For a function $f(x)$f(x) we say that the inverse function is $f^{-1}\left(x\right)$f1(x). Remember that inverse means to "undo", so from the output of $f(x)$f(x) we can figure out what the original input to $f(x)$f(x) was.

  $f$f   $f^{-1}$f1  
$a$a $\to$ $b$b $\to$ $a$a


Finding the inverse graphically

Notice that from the notation above, if $\left(a,b\right)$(a,b) is a point on the function $f(x)$f(x), then $\left(b,a\right)$(b,a) is a point on $f^{-1}\left(x\right)$f1(x).

The inverse can be found graphically by "swapping" the $x$x and $y$y coordinates for every single point on the function.

Did you know?

Swapping the $x$x-coordinate and the $y$y-coordinate for an ordered pair is the same as reflecting that point in the line $y=x$y=x.

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 $(3,1)$(