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.