Question

using the following equations, find g(f(x)): f(x) = -x² - 1 g(x) = x + 5 g(f(x)) = ?x² +

Explanation:

Step1: Substitute f(x) into g(x)

To find \( g(f(x)) \), we substitute \( f(x) = -x^2 - 1 \) into \( g(x) \). So \( g(f(x)) = g(-x^2 - 1) \).

Step2: Apply the function g(x)

Since \( g(x) = x + 5 \), we replace \( x \) in \( g(x) \) with \( -x^2 - 1 \). So \( g(-x^2 - 1) = (-x^2 - 1) + 5 \).

Step3: Simplify the expression

Simplify \( (-x^2 - 1) + 5 \). Combine the constant terms: \( -1 + 5 = 4 \). So the expression becomes \( -x^2 + 4 \).

Answer:

The coefficient of \( x^2 \) is \(-1\) and the constant term is \(4\), so \( g(f(x)) = \boxed{-1}x^2 + \boxed{4} \)