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Finding the Inverse of a Function
(KS4, Year 10)
The Lesson
An inverse function is a function that reverses another function. If we have a function f(x), we can find the inverse function f^{−1}(x).How to Find the Inverse of a Function
Finding the inverse of a function is easy.Question
Find the inverse function of the function below.StepbyStep:
1
Rearrange the function to find "x =".
We have rearranged f(x) = ½x + 1 to find what "x = ".
x = 2( f(x) − 1 )
f(x) = ½x + 1  
f(x) − 1 = ½x + 1 − 1  Subtract 1 from both sides 
f(x) − 1 = ½x  
2 × ( f(x) − 1 ) = 2 × ½x  Multiply both sides by 2 
2( f(x) − 1 ) = x  
x = 2( f(x) − 1 ) 
2
Replace x with f^{−1}(x).
f^{−1}(x) = 2( f(x) − 1 )
3
Replace f(x) with x.
f^{−1}(x) = 2( x − 1 )
Answer:
The inverse of the function f(x) = ½x + 1 is: f^{−1}(x) = 2(x − 1).Why Do We Relabel the Input and the Output?
When we find the inverse of a function, we replace: the input of the function (x) with the output of the inverse function (f^{−1}(x)), and
 the output of the function (f(x)) with the input of the inverse function (x).

An inverse function reverses a function, relating the function's output f(x) to its input x.
We can think of f(x) being the input to the inverse function and x being its output. 
An inverse function is a function. Using functional notation, an inverse function relates an input x to an output f^{−1}(x).
...we see that we need to replace f(x) with x and x with f^{−1}(x).
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