The Inverse Function

Pure Mathematics — Second Secondary

When does an inverse exist?

A function has an inverse only if it is one-to-one, meaning that different inputs always give different outputs.
If two inputs gave the same output, we could not tell which input to return when reversing the function.
The graphical test is the horizontal line test if any horizontal line meets the graph in more than one point, the function has no inverse.

Function	Has an inverse?	Reason
f(x) = 2x − 6	Yes	It is a linear increasing function
f(x) = x³ + 1	Yes	A horizontal line meets the graph only once
f(x) = x²	No (on all real numbers)	f(2) = f(−2) = 4

The Inverse Function · 2 / 7