A function, f, is the inverse of another , f−1, if:

f(a) = b, and f−1(b) = a.

Note that:

The domain f−1 is the range of f.

The range of f−1 is the domain of f.

To find the range of a function we have to find the domain of its inverse function.

If two functions are the inverse of each other, their composition is the identity function.

(f o f −1) (x) = (f −1 o f) (x) = x

The graphs of f and f −1 are symmetrical about the bisector of the first and third quadrant.

We must distinguish between the inverse function, f−1(x), and the inverse of a function, .

Calculation of the Inverse Function

Example 1.

Write the equation of the function with x and y.

Example 2.

Work out the value of the variable x as a function of the variable y.

Example 3.

The variables are exchanged.

Calculate the inverse function of:

Check the result for x = 2.

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