Question

If \( f(x) = \sqrt{x \) and \( g(x) = 2x - 3 \), then \( (f \circ g)(x) \) is:}

• A. \( (-\infty, -3) \)
• B. \( (-\infty, -3/2) \)
• C. \( [-3/2, 0] \)
• D. \( [0, 3/2] \)
• E. \( [3/2, \infty) \)

Hint:

When dealing with $\sqrt{g(x)}$, the domain is found by solving the inequality $g(x) \ge 0$.

Answer 1

Answer: (E)

Concept: The composition \( (f \circ g)(x) \) means \( f(g(x)) \). The domain of this composed function is restricted by the requirements of the outer function \( f \). Specifically, the output of \( g(x) \) must be a valid input for \( f(x) \).

Step 1: Form the composite function.
Substitute \( g(x) \) into \( f(x) \): \[ (f \circ g)(x) = f(2x - 3) = \sqrt{2x - 3} \]

Step 2: Identify the domain constraints.
For the square root function to be defined in real numbers, the expression inside must be non-negative: \[ 2x - 3 \ge 0 \]

Step 3: Solve for \( x \).
\[ 2x \ge 3 \] \[ x \ge \frac{3}{2} \] In interval notation, this is \( [3/2, \infty) \). Note: The question asks for \( (f \circ g)(x) \), but the options are provided in interval notation, implying it seeks the domain of the resulting function.