If f(x)=x^2-x+5, x12, and g(x) is its inverse function, then g'(7) equals:

Question

If $f(x)=x^2-x+5$, $x>\dfrac12$, and $g(x)$ is its inverse function, then $g'(7)$ equals:

Accepted Answer

$y=x^2-x+5\Rightarrow\left(x-\dfrac12\right)^2=y-\dfrac{19}{4}\Rightarrow x=\dfrac12+\sqrt{y-\frac{19}4}$ (taking $x>\frac12$).

$g(x)=\dfrac12+\sqrt{x-\frac{19}4}$, $g'(x)=\dfrac{1}{2\sqrt{x-19/4}}$.

$g'(7)=\dfrac{1}{2\sqrt{28/4-19/4}}=\dfrac{1}{2\sqrt{9/4}}=\dfrac{1}{2\cdot3/2}=\dfrac13$.

Solutions · 1