The inverse of a function is obtained by switching the x- and y-variables in the function. The inverse of function is written as f ^-1. The new relation obtained by reversing the x- and y-values is not necessarily a function itself. The new relation is only a function if the original function is a one-to-one function .

Shown below are three representations of the quadratic parent function.

The quadratic parent function (y = x²) is not a one-to-one function because elements of the range correspond to more than one element of the domain. For example, when y = 9 the domain is -3 and 3, or the range 9 corresponds to -3 and 3. Since it is not a one-to-one function, the inverse of this function is a relation.

Example

Equation

To find the inverse of this function from the equation, begin by exchanging the x- and y-variables in the equation and then rewrite the equation in y = form.

y = x²
x = y²
y² = x
√y² = √x
y = ± √x

Graph

• The graph of the inverse of a function is a reflection across the line y = x of the original function.
• The x-coordinate of a point on the original function becomes the y-coordinate of a corresponding point on the inverse.
• The y-coordinate of the same point on the original function becomes the x-coordinate of the corresponding point on the inverse.

Table

When given a table, the inverse of a function is found by reversing the x- and y-coordinates of the original function.