Question

How do you find the inverse of `f(x) = 4(x + 5)^2 - 6`?

Answer 1

No inverse function exists.

Explanation:

If there is no boundary on the domain, `f(x)` has no inverse function .

It would have been possible, however, if the was a boundary like `x >= -5`.

Why is this? Let me show you how to compute the inverse if there was one.

First, set `y = f(x)`:
`y = 4(x+5)^2 - 6`

Next, exchange the roles of `x` and `y`:
`x = 4(y+5)^2 - 6`

Now, try to solve this equation for `y` in terms of `x`:

... add `6` on both sides ...
`x + 6 = 4(y+5)^2`

... divide by `5` on both sides...
`(x+6)/4 = (y+5)^2`

Unfortunately there is no unique way to invert the square function since both the positive and the negative roots are solutions
(E.g. `x^2 = 9` has two solutions: `x = 3` and `x = -3` since both `3^2 = 9` and `(-3)^2 = 9` hold.)

This means that we would get something like:
`+- sqrt((x+6)/4) = y+5`

And finally, adding `-5` on both sides gives us:
`y = -5 +- sqrt ((x+6)/4)`

However, this means that for a unique value of `x` you don't get a unique value of `y` but two different values. This means that this is no function and thus, your problem has no solution.