2.3.2.2 Inverse Functions

Identity Function

Function that maps each value to itself

• id(x) = x

Inverse Function

An inverse function reverses the effect of a function

Inverse Function Notation

Conditions for an Inverse Function

• Only one-to-one functions have inverses

• A function has an inverse if its graph passes the horizontal line test: Any horizontal line will intersect with the graph at most once

Where Composite Functions are Inverses

If ‘f o g’ and ‘g o f’ have the same effect as the identity function then ‘f’ and ‘g’ are inverses

Where a Compose Function is Composed of its Inverses

Domain of an Inverse Function

The range of a function becomes the domain of its inverse

Range of an Inverse Function

The domain of a function becomes the range of its inverse

Graphs of the Inverse Function

• The inverse function ‘y = f^-1(x)’ is just the mirror image of the original function ‘y = f(x)’ across the line y = x

• If f(x) intersects y = x, then f(x) and f^-1(x) intersect there too

  • Because reflecting that point across y = x does nothing — it stays the same point

  • But the function and its reflection could also cross somewhere else, not just on the mirror line

Finding the Inverse of a Function

Inverse Function of a Many-to-One Function

• Rrestrict the domain of a many-to-one function so that it has an inverse

• Choose a subset of the domain where the function is one-to-one

  • The inverse will be determined by the restricted domain

  • Note that a many-to-one function can only have an inverse if its domain is restricted first

Inverse Function of a Many-to-One Function: Quadratic Function

Inverse Function of a Many-to-One Function: Trigonometric Function

Finding the Inverse Function of a Many-to-One Function

• The restricted domain (from a many-to-one to one-to-one function) of the original function is the same as the range of the inverse function

• Hence after solving for the inverse function, the range of this inverse function is restricted in the same fashion that the domain of the original function is