Question

If \(\mathrm{f}(x)=3 x+10, \mathrm{~g}(x)=x^2-1\), then \((\mathrm{fog})^{-1}(x)=\)

• A. \((\frac{x-7}{3})\)
• B. \((\frac{x-7}{3})^{\frac{1}{2}}\)
• C. \((\frac{x-7}{3})^{\frac{1}{3}}\)
• D. \((\frac{3}{x-7})^{\frac{3}{2}}\)

Hint:

Form fog(x), then solve y = fog(x) for x.

Answer 1

The Correct Option is B

Step 1: Rule
Find the composition first and then invert it.

Step 2: Invert
From $3x^2 + 7 = y$, the inverse function is $x = \sqrt{\dfrac{y-7}{3}}$.

Step 3: Check
Substituting back: $3\cdot\dfrac{y-7}{3} + 7 = y$, so the answer is consistent.

Final Answer:
Option B is the inverse. \[ \boxed{\text{(B) }\left(\dfrac{x-7}{3}\right)^{\frac{1}{2}}} \]