Finding the inverse function of a quadratic function

Question

let $f:[-4,∞) \rightarrow \mathbb{R} , f(x)=-(x+4)^2 +3$. show that $f^{-1}:(-∞,3] \rightarrow \mathbb{R}, f^{-1}(x)=\sqrt{3-x}-4.$

a question from my 11th-grade maths assignment. I don't even know where to start. please help.

Answer 1

Just follow this procedure when trying to calculate inverse functions:

STEP 1:

Write the initial function

$$ f(x) = -(x+4)^2+3 $$

as

$$ y = -(x+4)^2+3 $$

STEP 2:

Replace $y$ with $x$ like so

$$ x=-(y+4)^2+3 $$

STEP 3:

Solve the above in order to separate $y$

$$ \sqrt{3-x} -4 = y $$

STEP 4:

Replace this new $y$ with $f^{-1}(x)$

$$ \sqrt{3-x} -4 = f^{-1}(x) $$

Find the domain of this new function. Can you proceed?

STEP 5:

Check your work knowing that $$f^{-1}(f(x))=x$$

If you do some algebra, verify that your result (in this case)

$$ f^{-1}(f(x)) = \sqrt{3-f(x)} -4 = \sqrt{3-(-(x+4)^2+3)} -4 = x $$

Answer 2

Hint:

For $x\ge-4$, solve the equation $-(x+4)^2+3=y$ and find $x$ in term of $y$.