1. ## Inverse function

How can you find the inverse $f^{-1}(x)$ of $f(x)=x^2+3x$ by restricted the domain?

I looked at this and expected to end up with a square root of something, so that you could restrict the domain to only include positive or negative values, but then I tried to solve $f(f^{-1}(x))=x$, and couldn't. Can anyone help?

2. Hello,
Originally Posted by Greengoblin
How can you find the inverse $f^{-1}(x)$ of $f(x)=x^2+3x$ by restricted the domain?

I looked at this and expected to end up with a square root of something, so that you could restrict the domain to only include positive or negative values, but then I tried to solve $f(f^{-1}(x))=x$, and couldn't. Can anyone help?
I don't know if it stands for any equation of this type, but it's worth the try.

$y=x^2+3x$

Complete the square on the RHS :

$x^2+3x=x^2+2 \cdot {\color{red}\frac 32} \cdot x+\left({\color{red}\frac 32}\right)^2-\left({\color{red}\frac 32}\right)^2=\left(x+\frac 32\right)^2-\frac 94$

Therefore $y=\left(x+\frac 32\right)^2-\frac 94$

$y+\frac 94=\left(x+\frac 32\right)^2$

Isolate x And as you said, restrict the domain !

3. Thanks, but I'm not sure what you've done there. Is supposed to be the inverse function? If so, I'm not sure how you got to it. I was told I need to solve the equation $f(f^{-1}(x))=x$, for these types of questions.

4. Originally Posted by Greengoblin
Thanks, but I'm not sure what you've done there. Is supposed to be the inverse function? If so, I'm not sure how you got to it. I was told I need to solve the equation $f(f^{-1}(x))=x$, for these types of questions.
Nope, the inverse function is in the form x=...

Some people invert at the beginning x and y, so that it is clearer...

Find the expression of x with respect to y and this will give you the inverse function :

$x=f^{-1}(y)$

5. Ok, I think I can see what you've done....I was having trouble simplifying:

$f(f^{-1}(x))=x$
$(f^{-1}(x))^2+3(f^{-1}(x))=x$

and didn't recognize the completing the square teccnique.

I think I can do it now, thanks alot!