notes/education/math/MATH1050/Inverse Functions.md

For a function to have an inverse, it needs to have one x for every y, and vice versa. You can use the horizontal line test to verify that the inverse of a function is valid. If you can draw a horizontal line and it crosses through two points at the same time at any height, the inverse is not a valid function. To get the inverse, you can switch the x and y of a function, and it will mirror the graph over the line y = x.

Examples

Given the below function:

y = \frac{1}{2}x + 3

You can find the inverse by switching the x and y values and solving for y:

x = \frac{1}{2}y + 3

The range of the inverse is the same as the domain of the original. You can verify by taking f \circ g, and simplifying.