Question

How do you find the inverse of `y = - |x-3| + 5`?

Answer 1

`f^(-1)(x) = 3 pm (5 - x)`

Explanation:

`y = -|x-3| + 5`

`|x-3| = 5 - y`

`x-3 = pm (5 - y)`

`x = 3 pm (5 - y)`

The inverse function is therefore:

`f^(-1)(x) = 3 pm (5 - x)`

There would be no harm in verifying this for `x = 2` and `x = 4`, ie either side of `x = 3` which is the point at which `(x-3)` changes sign.

For `x = color(red)(2)`, we get `y = 4`. Our inverse function suggests that `f^(-1)(4) = 3 pm 1 = 4, color(red)(2)`.

And for `x = color(blue)(4)`, we also get `y = 4`, which we expect from the symmetry. Our inverse function suggests that `f^(-1)(4) = 3 pm 1 = color(blue)(4),2`.

graph{y = -|x-3| + 5 [-10, 10, -5, 5]}