Question

Find the inverse function of 𝑓(𝑥) = −𝑥² + 8𝑥 − 14 if 𝑥 ≤ 4. Find the domain of this inverse function.

Answer 1

$$f(x)-x^2+8x-14=-(x^2-8x+16)+2$$

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

The function $f(x)=-x^2+8x-14, x\leq4$ is one-to-one. Then it has inverse.

Replace $f(x)$ with $y$

$$y=-(x-4)^2+2, x\leq4, y\leq2$$

 Replace every $x$ with a $y$ and replace every $y$  with an $x$

$$x=-(y-4)^2+2, x\leq2, y\leq4$$

Solve for $y$

$$(y-4)^2=2-x, x\leq2, y\leq4$$

$$y-4=-\sqrt{2-x}$$

$$y=4-\sqrt{2-x}$$

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

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

Domain $(-\infin, 2]$