Question

How do you graph the function `f(x)=absx+2` and its inverse?

Answer 1

Use the parent function: `f(x) = |x|` and move the vertex to `(0,2)`

Explanation:

The parent function `f(x) = |x|` looks like:

graph{|x| [-9.06, 8.72, -2.36, 6.53]}

Move the vertex to `(0,2)`:

graph{|x|+2 [-10.17, 9.83, -2.916, 7.085]}

or do point-plotting: `(-3,5), (-1, 3), (0, 2), (1, 3), (3, 5), (5, 7)...`

The inverse can be created a number of ways.
One way is to reverse the `x` and `y`'s and plot the reversed points. But you need to realize in order to have an inverse, the original function must be limited in its' domain. To be a function of `x` the function must pass both the vertical and horizontal line test. To pass the horizontal line test, the original function must limit its' domain to `x >= 0`. This means the `y` values of the inverse function must be limited to `y>=0`
Inverse function's points for point plotting: `(2, 0), (3, 1), (5, 3), (7, 5), ...`

A 2nd way to find the inverse function is to
1. Make `f(x) = y`: `y = |x| + 2`
2. Switch `x` and `y`: `x = |y| + 2`
3. Solve for `y: |y| =x - 2`
4. For `y >= 0: y = x - 2`

Note: The inverse function is always mirrored about the `y = x` axis.