# Inverse functions

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##### Intros
###### Lessons

1. • What is "inverse", and what does "inverse" do to a function?
• Inverse: switch "x" and "y"
• Inverse: reflect the original function in the line "y = x"
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##### Examples
###### Lessons
1. Graph an inverse
Given the graph of $y = f\left( x \right)$ as shown,
1. 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 $f\left( x \right)$
2. Is $f\left( x \right)$ a function?
Is ${f^{ - 1}}\left( x \right)$ a function?
2. Inverse of a Quadratic Function
Consider the quadratic function: $f(x) = (x+4)^2 + 2$
1. Graph the function $f\left( x \right)$ and state the domain and range.
2. Graph the inverse ${f^{ - 1}}\left( x \right)$ and state the domain and range.
3. 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.
3. Determine the equation of the inverse.
Algebraically determine the equation of the inverse ${f^{ - 1}}\left( x \right)$, given:
1. $f\left( x \right) = - 5x + 4$
2. $f\left( x \right) = {\left( {7x - 8} \right)^3} - 1$
3. $f\left( x \right) = \frac{{3x}}{{2 + x}}$
###### Topic Notes
An inverse function is a function that reverses all the operations of another function. Therefore, an inverse function has all the points of another function, except that the x and y values are reversed.