Question

Let \( f(x) = \log_e(x) \) and let \( g(x) = \frac{x - 2}{x^2 + 1} \). Then the domain of the composite function \( f \circ g \) is:

• A. \( (2, \infty) \)
• B. \( (-1, \infty) \)
• C. \( (0, \infty) \)
• D. \( (1, \infty) \)
• E. \( (1, 0) \)

Hint:

When dealing with composite functions, remember that the domain of the composite function is determined by the domain restrictions of both functions involved.

Answer 1

The Correct Option is A

The composite function \( f \circ g \) is defined as \( f(g(x)) \).
The function \( f(x) = \log_e(x) \) is defined for all \( x>0 \), so for \( f(g(x)) \) to be defined, we need \( g(x)>0 \).
Next, consider the function \( g(x) = \frac{x - 2}{x^2 + 1} \). We need to find when \( g(x)>0 \).
\[ g(x) = \frac{x - 2}{x^2 + 1}>0 \] Since \( x^2 + 1>0 \) for all real \( x \), the inequality holds when \( x - 2>0 \), which simplifies to: \[ x>2 \] Thus, for \( f(g(x)) \) to be defined, \( x>2 \).
Therefore, the domain of the composite function \( f \circ g \) is \( (2, \infty) \).