Determine the inverse function.

[Chocolate]

f(x)=(x+3)^2

My work:

y=(x+3)(x+3)
y=x^2+3x+3x+9
y=x^2+6x+9
x=y^2+6y+9
y^2+6y=x-9

I can't isolate y.

 

[soroban]

Hello, Chocolate!

Find the inverse function of: \(\displaystyle f(x) \,= \,(x\,+\,3)^2\)

We have: . \(\displaystyle (x\,+\,3)^2 \,= \,y\)

Then: . \(\displaystyle (y\,+\,3)^2 \,= \,x\)

Take square roots: . \(\displaystyle y\,+\,3 \,= \,\pm\sqrt{x}\)

Hence: . \(\displaystyle y \:= \:-3\,\pm\,\sqrt{x}\)

Therefore: . \(\displaystyle f^{-1}(x) \:= \:-3\,\pm\,\sqrt{x}\)

. . (Note that the inverse is not a function.)