Question

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

Answer 1

No inverse function exists without domain restrictions.

Explanation:

Set `f(x) = y`:

`y = (x+2)^2 - 4`

Interchange `y` and `x` in your equation:

`x = (y+2)^2 - 4`

Now, you need to solve this equation for `y`.
First of all, add `4` on both sides:

`x + 4 = (y+2)^2`

The next step would be to draw the root. However, this will leave you with two solutions, since e.g. for `25 = x^2`, both `5 = x` and `-5 = x` are solutions.

`sqrt(x+4) = abs(y+2)`

`<=> +-sqrt(x + 4) = y + 2`

Subtract `2` on both sides:

`-2 +- sqrt(x+4) = y`

Beware that a function must have a unique value for `y` for each unique value of `x`.

However, this is not the case here since for e.g. `x = 12`, you have both `y = - 2 + sqrt(16) = 2` and `y = - 2 - sqrt(16) = -6`.

This means that there no inverse function exists.

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Remark:

An inverse function would exist if you restricted the domain of the original function.

As you can easily see that the vertex of the function is at `x = -2`, it would suffice to either restrain the domain to e.g. `x <= -2` or to `x >=-2`.

For example, if your original function was

`f(x) = (x+2)^2 -4 " where " x >=-2`

then you could continue with the calculation from above, abandoning the negative term:

`y = -2 + sqrt(x+4)`

Replace `y` with `f^(-1)(x)`:

`f^(-1)(x) = -2 + sqrt(x+4)`