Question

If \( f(x) = x^3 - x \) and \( g(x) = \sin(2x) \), then \( f(g\left(\frac{\pi}{12}\right)) \) is:

• A. \( 0 \)
• B. \( 1 \)
• C. \( \frac{-3}{8} \)
• D. \( 2 \)

Hint:

Start by evaluating the inner function and then substitute into the outer function. Be cautious with arithmetic.

Answer 1

The Correct Option is C

We are given \( f(x) = x^3 - x \) and \( g(x) = \sin(2x) \). We need to find \( f(g\left(\frac{\pi}{12}\right)) \).
Step 1: Find \( g\left(\frac{\pi}{12}\right) \). \[ g\left(\frac{\pi}{12}\right) = \sin\left(2 \times \frac{\pi}{12}\right) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
Step 2: Substitute into \( f(x) \). \[ f\left(g\left(\frac{\pi}{12}\right)\right) = f\left(\frac{1}{2}\right) = \left(\frac{1}{2}\right)^3 - \frac{1}{2} = \frac{1}{8} - \frac{4}{8} = \frac{-3}{8} \]