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

##### 1 Answer

No inverse function exists without domain restrictions.

#### Explanation:

Set

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

Interchange

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

Now, you need to solve this equation for

First of all, add

#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

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

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

Subtract

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

Beware that a function must have a **unique** value for

However, this is not the case here since for e.g.

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

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

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