Sure, let's break down the solution step by step for finding [tex]\((g - f)(3)\)[/tex].
First, we are given the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ f(x) = 4 - x^2 \][/tex]
[tex]\[ g(x) = 6x \][/tex]
We are required to find [tex]\( (g - f)(3) \)[/tex], which means we need to evaluate both [tex]\( g(3) \)[/tex] and [tex]\( f(3) \)[/tex], and then subtract [tex]\( f(3) \)[/tex] from [tex]\( g(3) \)[/tex].
1. Evaluate [tex]\( f(3) \)[/tex]:
[tex]\[ f(x) = 4 - x^2 \][/tex]
[tex]\[ f(3) = 4 - 3^2 \][/tex]
[tex]\[ 3^2 = 9 \][/tex]
[tex]\[ f(3) = 4 - 9 \][/tex]
[tex]\[ f(3) = -5 \][/tex]
2. Evaluate [tex]\( g(3) \)[/tex]:
[tex]\[ g(x) = 6x \][/tex]
[tex]\[ g(3) = 6 \cdot 3 \][/tex]
[tex]\[ g(3) = 18 \][/tex]
3. Compute [tex]\( (g - f)(3) \)[/tex]:
[tex]\[ (g - f)(3) = g(3) - f(3) \][/tex]
[tex]\[ (g - f)(3) = 18 - (-5) \][/tex]
[tex]\[ (g - f)(3) = 18 + 5 \][/tex]
[tex]\[ (g - f)(3) = 23 \][/tex]
Thus, the expression that is equivalent to [tex]\((g - f)(3)\)[/tex] evaluates to 23.
