Question

Let \( f(x) = \log x \) and \[ g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \] Then the domain of \( f \circ g \) is:

• A. \( \mathbb{R} \)
• B. \( (0, \infty) \)
• C. \( [0, \infty) \)
• D. \( [1, \infty) \)

Hint:

For composite functions, analyze the inner function's range and ensure it aligns with the domain of the outer function.

Answer 1

The Correct Option is A

Step 1: Understanding domain constraints. The function \( f(x) = \log x \) requires \( x>0 \), so we must ensure \( g(x)>0 \) for \( f(g(x)) \) to be defined. 

Step 2: Finding domain of \( g(x) \). Given: \[ g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \] The denominator is a quadratic equation: \[ 2x^2 - 2x + 1 \] Since the discriminant is negative, it is always positive. 

Step 3: Solving for \( g(x)>0 \). Setting the numerator \( x^4 - 2x^3 + 3x^2 - 2x + 2>0 \), we find that \( x<0 \) satisfies this condition. Hence, \( g(x) \) is always positive. Thus, \( g(x)>0 \) for all \( x \), meaning the domain of \( f \circ g \) is \( \mathbb{R} \).