How to Find the Inverse of a Function Using Algebra (Mathematics Lesson)
How to Find the Inverse of a Function Using AlgebraThe inverse of a function can be found by rearranging the function using algebra and then relabelling the input and output.
A Real Example of How to Find the Inverse of a Function Using AlgebraQuestion: What is the inverse function of the function:
Step 1: Rearrange the function so the input x is the subject of the equation.
Initially, the output is written in terms of the input, f(x) =
Rearrange the equation using algebra (in this case algebra with addition).
The input is now written in terms of the output, x =
Step 2: Relabel the function so the input becomes the output and the output becomes the input.
- The old input x becomes our new output f-1(x)
x → f-1(x)
- The old output f(x) becomes our new input x.
f(x) → x
The inverse function is
Another Real Example of How to Find the Inverse of a Function Using AlgebraThe slider below shows a real example of how to find the inverse of a function using algebra.
NoteWHAT IS AN INVERSE FUNCTION?
An inverse function is itself a function which reverses a function.
If a function f(x) maps an input x to an output f(x)...
... an inverse function takes the output f(x) back to the input x:
An inverse function is denoted f-1(x). It relates an input x to an output f-1(x):
WHY DO WE RELABEL THE INPUT AND OUTPUT?
In Step 2 (see left), we relabel the input as the output and the output as the input. Why is this?
Consider the two arrow diagrams of f-1(x) shown in the note above, (with the first swapped around so the input is on the left):
- The output of the function f(x) is equivalent to the input of the inverse function x.
- The input of the function x is equivalent to the output of the inverse function f-1(x).