To find [tex]\( g(f(x)) \)[/tex] for the given functions [tex]\( f(x) = x - 7 \)[/tex] and [tex]\( g(x) = 5x + 2 \)[/tex], we can follow these steps:
1. Find [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = x - 7
\][/tex]
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex] to find [tex]\( g(f(x)) \)[/tex]:
Because [tex]\( g(x) \)[/tex] is given by [tex]\( g(x) = 5x + 2 \)[/tex], we substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[
g(f(x)) = g(x - 7)
\][/tex]
3. Apply the function [tex]\( g \)[/tex] to [tex]\( (x - 7) \)[/tex]:
Substitute [tex]\( (x - 7) \)[/tex] into the expression for [tex]\( g(x) \)[/tex]:
[tex]\[
g(x - 7) = 5(x - 7) + 2
\][/tex]
4. Simplify the expression:
[tex]\[
g(x - 7) = 5(x - 7) + 2 = 5x - 35 + 2
\][/tex]
[tex]\[
g(x - 7) = 5x - 33
\][/tex]
Thus, the function [tex]\( g(f(x)) \)[/tex] can be written in the form:
[tex]\[ g(f(x)) = 5x - 33 \][/tex]
From this expression, we see that the coefficient of [tex]\( x \)[/tex] is [tex]\( 5 \)[/tex] and the constant term is [tex]\( -33 \)[/tex]. Therefore:
[tex]\[ g(f(x)) = 5x + (-33) \][/tex]
So, the coefficients are:
[tex]\[ g(f(x)) = 5x + (-33) \][/tex]
The coefficient of [tex]\( x \)[/tex] is [tex]\( 5 \)[/tex].
The constant term is [tex]\( -33 \)[/tex].
Thus, the function [tex]\( g(f(x)) \)[/tex] can be expressed as:
[tex]\[ g(f(x)) = 5x - 33 \][/tex]
