Given \( f(x)=\frac{5}{2 x+1} \) and \( g(x)=x^{2}-3 \) Olivia was asked to find \( f(g(x)) \). She showed the following work: Line 1: \( f(g(x))=f\left(x^{2}-3\right) \) Line 2: \( \quad=\frac{5}{2 x^{2}-3+1} \) Line 3: \( \quad \frac{5}{2 x^{2}-2} \) This answer is incorrect. Answer the following questions: A. On which line did Olivia make a mistake? Justify your answer. (5 points)

Olivia made a mistake on Line 2. In Line 1, she correctly identified that \( f(g(x)) = f(x^2 - 3) \). However, in Line 2, she incorrectly substituted \( g(x) \) into \( f(x) \). The function \( f(x) \) is defined as \( f(x) = \frac{5}{2x + 1} \). When substituting \( g(x) = x^2 - 3 \) into \( f(x) \), we should replace \( x \) in \( f(x) \) with \( g(x) \): \[ f(g(x)) = f(x^2 - 3) = \frac{5}{2(x^2 - 3) + 1} \] Instead, Olivia wrote \( 2x^2 - 3 + 1 \), which is incorrect. The correct substitution should be: \[ 2(x^2 - 3) + 1 = 2x^2 - 6 + 1 = 2x^2 - 5 \] Thus, the correct expression for Line 2 should be: \[ f(g(x)) = \frac{5}{2(x^2 - 3) + 1} = \frac{5}{2x^2 - 5} \] Therefore, the mistake occurred in Line 2 when she incorrectly simplified the expression.