Question

How do you determine whether the function `q(x)=(x-5)^2` has an inverse and if it does, how do you find the inverse function?

Answer 1

No inverse function

Explanation:

The function can be rewritten as an equation: `y=(x-5)^2`. To find the inverse function, we solve for `x` in terms of `y`.

`y=(x-5)^2`
`+-sqrt(y)=x-5`
`x=+-sqrt(y)+5`

The inverse function of `q(x)` then should be `q^-1(x)=+-sqrt(x)+5` (we replaced the `y` in the equation for `x`). However, remember that, for functions, for any value `x`, there is only one value `y` for that `x`. This is clearly not the case here, since, for most values of `x` (except `x=0`), there are two possible `y` values because of the `+-` sign.

Therefore, there are no inverse functions for `q(x)`.