Inverse Functions
(KS4, Year 10)

An inverse function is a function that reverses another function. If a function f relates an input x to an output f(x)...

...an inverse function f⁻¹ relates the output f(x) back to the input x:

Imagine a function f relates an input 2 to an output 3... ...the inverse function f⁻¹ relates 3 back to 2... An inverse function is denoted f⁻¹(x).

A Real Example of an Inverse Function

It is easier to understand inverse functions with an example. Let us consider a function:

f(x) = x + 1 and Its Inverse

The function f(x) = x + 1 takes each input and adds 1 to it. The mapping diagram below shows this function.

Now let's consider the inverse function of f(x) = x + 1. The inverse function will take us back to the original values. The inverse function takes each input and subtracts 1 from it. The inverse of f(x) = x + 1 is:

The Inverse Function Takes Us Back to the Input

We can combine a function with its inverse.

• Imagine we put a value into a function and then put the result into the inverse function. The output is original value.
  
  If we put an input x into a function f(x), and then put the result into the inverse function f⁻¹(x), the output is x.
  
  
• Imagine we put a value into the inverse function and then put the result into the function. The output is original value.
  
  If we put an input x into a function f(x), and then put the result into the inverse function f⁻¹(x), the output is x.
  
  

Either way, if we combine a function with its inverse, we get back where we started.

Graphs of Inverse Functions

Functions and their inverses can be plotted on graphs.

• Functions relate an input x to an output y. We can plot these pairs of inputs and outputs as Cartesian coordinates, (x, y).
• The inverse function relates the output y back to the input x. We swap the coordinates around to find the inverse function, such that (x, y) becomes (y, x).

The inverse function can be found by reflecting the graph of the function in the line y = x.

The Horizontal Line Test

For the inverse function to be a function, each input can only be related to one output. This means that for the function (which will be reflected in y = x), each value of y can only be related to one value of x. On a graph, this means that any horizontal line only crosses the curve once. This is the horizontal line test. The curve below passes the horizontal line test. Any horizontal line that is drawn will cross the curve only once. The inverse will be a function:The curve below does not pass the horizontal line test. A horizontal line can be drawn which crosses the curve twice. The inverse will not be a function:

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