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

1 Answer
Lotusbluete
Jan 25, 2016

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=x2, both 5=x and −5=x are solutions.

√x+4=|y+2|

⇔±√x+4=y+2

Subtract 2 on both sides:

−2±√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+√16=2 and y=−2−√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+√x+4

Replace y with f−1(x):

f−1(x)=−2+√x+4

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