To find the inverse function of [tex]\( f(x) = 9x^2 - 12 \)[/tex] for [tex]\( x \geq 0 \)[/tex], we need to follow a series of steps to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex], and then swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex] to get the inverse function.
1. Start with the given function
[tex]\[
y = 9x^2 - 12
\][/tex]
2. Add 12 to both sides of the equation to isolate the term involving [tex]\( x \)[/tex]
[tex]\[
y + 12 = 9x^2
\][/tex]
3. Divide both sides by 9 to express [tex]\( x^2 \)[/tex] in terms of [tex]\( y \)[/tex]
[tex]\[
\frac{y + 12}{9} = x^2
\][/tex]
4. Take the square root of both sides to solve for [tex]\( x \)[/tex]. Since the domain is restricted to [tex]\( x \geq 0 \)[/tex], we only take the positive root.
[tex]\[
x = \sqrt{\frac{y + 12}{9}}
\][/tex]
5. To express the inverse function, swap [tex]\( x \)[/tex] and [tex]\( y \)[/tex]
[tex]\[
f^{-1}(x) = \sqrt{\frac{x + 12}{9}}
\][/tex]
This means that the inverse function is
[tex]\[
f^{-1}(x) = \frac{\sqrt{x + 12}}{3}
\][/tex]
Looking at the given options:
A. [tex]\(h(x)=\frac{\sqrt{x-12}}{3}\)[/tex]
B. [tex]\(g(x)=\frac{\sqrt{x+12}}{3}\)[/tex]
C. [tex]\(p(x)=\frac{\sqrt{x-12}}{9}\)[/tex]
D. [tex]\(q(x)=\frac{\sqrt{x+12}}{}\)[/tex]
The correct inverse function that matches our derived form [tex]\( \frac{\sqrt{x + 12}}{3} \)[/tex] is option B.
Therefore, the correct answer is:
[tex]\[
\boxed{2 \text (This corresponds to option B)}
\][/tex]
