Question

Find the domain of the composite function \( f \circ g(x) \) where \( f(x) = \log(5x) \) and \( g(x) = \cos(x) \).

• A. \( \left[ 0, \frac{\pi}{2} \right] \)
• B. \( (-\infty, \infty) \)
• C. \( \left(0, \infty \right) \)
• D. \( \left( \cos^{-1} \left( \frac{1}{5} \right), \infty \right) \)

Hint:

When dealing with logarithmic functions, always ensure that the argument of the log is positive. In this case, find where \( \cos(x) \) is positive to determine the domain.

Answer 1

The Correct Option is C

We are asked to find the domain of the composite function \( f(g(x)) = \log(5 \cdot \cos(x)) \).
1. For the logarithmic function \( f(x) = \log(5x) \), the domain is \( x>0 \), meaning \( 5 \cdot \cos(x)>0 \). Therefore, we need: \[ \cos(x)>0 \]
2. The cosine function is positive for values of \( x \) in the intervals: \[ x \in \left( 0, \frac{\pi}{2} \right) \cup \left( \frac{3\pi}{2}, 2\pi \right) \dots \] Therefore, the domain of \( f(g(x)) \) is \( \left( 0, \infty \right) \).