Question

let $f(x)=2(x-5)^2+3$
find the inverse.
$f(x)^{-1} = \square$
graphing calculator

Explanation:

Step1: Replace $f(x)$ with $y$

$y = 2(x-5)^2 + 3$

Step2: Swap $x$ and $y$

$x = 2(y-5)^2 + 3$

Step3: Isolate the squared term

$x - 3 = 2(y-5)^2$
$\frac{x-3}{2} = (y-5)^2$

Step4: Solve for $y$

$y - 5 = \pm\sqrt{\frac{x-3}{2}}$
$y = 5 \pm\sqrt{\frac{x-3}{2}}$

Step5: Rewrite as inverse function

$f^{-1}(x) = 5 \pm\sqrt{\frac{x-3}{2}}$

Answer:

$\boldsymbol{f^{-1}(x) = 5 \pm\sqrt{\frac{x-3}{2}}}$