Question

Let \( f: \mathbb{R} \to \mathbb{R}, g: \mathbb{R} \to \mathbb{R} \) be two functions such that \[ f(x) = 2x - 3, \quad g(x) = x^3 + 5. \] \text{The function} \( (f \circ g)^{-1}(x) \) is equal to:

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

Hint:

To find the inverse of a composite function, first find the composition and then solve for the inverse.

Answer 1

The Correct Option is B

First, solve for \( (f \circ g)(x) \) and then take the inverse. The inverse function is \( \left( \frac{x - 7}{2} \right)^{1/3} \).
Final Answer: \[ \boxed{\left( \frac{x - 7}{2} \right)^{1/3}} \]