# Inverse Functions: Unraveling Mathematical Relationships Dive into the world of inverse functions! Discover how to graph, analyze, and apply these powerful mathematical tools. Perfect for students looking to enhance their algebra and pre-calculus skills.

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Examples

**Graph an inverse**

Given the graph of $y = f\left( x \right)$ as shown,

- Sketch the graph of the inverse $y = {f^{ - 1}}\left( x \right)$
in 2 ways:

i) by reflecting $f\left( x \right)$ in the line $y = x$

ii) by switching the x and y coordinates for each point on $f\left( x \right)$ - Is $f\left( x \right)$ a function?

Is ${f^{ - 1}}\left( x \right)$ a function?

- Sketch the graph of the inverse $y = {f^{ - 1}}\left( x \right)$
in 2 ways:
**Inverse of a Quadratic Function**

Consider the quadratic function: $f(x) = (x+4)^2 + 2$- Graph the function $f\left( x \right)$ and state the domain and range.
- Graph the inverse ${f^{ - 1}}\left( x \right)$ and state the domain and range.
- Is ${f^{ - 1}}\left( x \right)$ a function?

If not, describe how to restrict the domain of $f\left( x \right)$ so that the inverse of $f\left( x \right)$ can be a function.

**Determine the equation of the inverse.**

Algebraically determine the equation of the inverse ${f^{ - 1}}\left( x \right)$, given: