select the correct answer. which function is the inverse of $f(x)=\frac{sqrt{x - 2}}{6}$?\na…

select the correct answer. which function is the inverse of $f(x)=\frac{sqrt{x - 2}}{6}$?\na. $f^{-1}(x)=36x^{2}+2$, for $xgeq0$\nb. $f^{-1}(x)=6x^{2}+2$, for $xgeq0$\nc. $f^{-1}(x)=36x + 2$, for $xgeq0$\nd. $f^{-1}(x)=6x^{2}-2$, for $xgeq0$

select the correct answer. which function is the inverse of $f(x)=\frac{sqrt{x - 2}}{6}$?\na. $f^{-1}(x)=36x^{2}+2$, for $xgeq0$\nb. $f^{-1}(x)=6x^{2}+2$, for $xgeq0$\nc. $f^{-1}(x)=36x + 2$, for $xgeq0$\nd. $f^{-1}(x)=6x^{2}-2$, for $xgeq0$

Answer

Explanation:

Step1: Let $y = f(x)$

$y=\frac{\sqrt{x - 2}}{6}$

Step2: Swap $x$ and $y$

$x=\frac{\sqrt{y - 2}}{6}$

Step3: Solve for $y$

First, multiply both sides by 6: $6x=\sqrt{y - 2}$. Then square both sides: $(6x)^2=y - 2$, which simplifies to $36x^{2}=y - 2$. Finally, add 2 to both sides: $y = 36x^{2}+2$. Since the original function $f(x)=\frac{\sqrt{x - 2}}{6}$ has a domain $x\geq2$ and range $y\geq0$, the inverse function $f^{-1}(x)$ has a domain $x\geq0$.

Answer:

A. $f^{-1}(x)=36x^{2}+2$, for $x\geq0$