Question

If \( f(x) = 4x^3 - 8 \), then what is the value of \( f^{-1}(-8) + f^{-1}(24) \)?

• A. -1
• B. 1
• C. 2
• D. 3

Hint:

To evaluate inverse function expressions, solve the original function algebraically and then substitute specific values.

Answer 1

The Correct Option is B

We solve \( f(x) = y \Rightarrow 4x^3 - 8 = y \Rightarrow x^3 = \frac{y + 8}{4} \Rightarrow x = \left( \frac{y + 8}{4} \right)^{1/3} \). Now, \[ f^{-1}(-8) = \left( \frac{-8 + 8}{4} \right)^{1/3} = 0
f^{-1}(24) = \left( \frac{24 + 8}{4} \right)^{1/3} = \left( \frac{32}{4} \right)^{1/3} = 2
\Rightarrow f^{-1}(-8) + f^{-1}(24) = 0 + 2 = 2 \] Correct value: 2