To find [tex]\((g - f)(7)\)[/tex], let's take the following steps:
1. Define the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ f(x) = 3x + 6 \][/tex]
[tex]\[ g(x) = x^2 + 3x - 6 \][/tex]
2. Evaluate [tex]\( f(7) \)[/tex]:
Substitute [tex]\( x = 7 \)[/tex] into the function [tex]\( f \)[/tex]:
[tex]\[ f(7) = 3(7) + 6 = 21 + 6 = 27 \][/tex]
3. Evaluate [tex]\( g(7) \)[/tex]:
Substitute [tex]\( x = 7 \)[/tex] into the function [tex]\( g \)[/tex]:
[tex]\[ g(7) = 7^2 + 3(7) - 6 \][/tex]
[tex]\[ g(7) = 49 + 21 - 6 \][/tex]
[tex]\[ g(7) = 70 - 6 = 64 \][/tex]
4. Calculate [tex]\((g - f)(7)\)[/tex]:
We need to find the difference [tex]\( g(7) - f(7) \)[/tex]:
[tex]\[ (g - f)(7) = g(7) - f(7) \][/tex]
[tex]\[ (g - f)(7) = 64 - 27 = 37 \][/tex]
So, the result of [tex]\((g - f)(7)\)[/tex] is:
[tex]\[
(g - f)(7) = 37
\][/tex]
Let's summarize the evaluated values:
- [tex]\( g(7) = 64 \)[/tex]
- [tex]\( f(7) = 27 \)[/tex]
- [tex]\( (g - f)(7) = 37 \)[/tex]
Therefore, the simplified result of [tex]\((g - f)(7)\)[/tex] is [tex]\( 37 \)[/tex].
