To find [tex]\( g(f(x)) \)[/tex] given the functions:
[tex]\[ f(x) = -7x + 1 \][/tex]
[tex]\[ g(x) = -5x - 6 \][/tex]
we need to perform the composition [tex]\( g(f(x)) \)[/tex]. This means we will substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex].
1. First, we determine [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = -7x + 1 \][/tex]
2. Next, we substitute [tex]\( f(x) \)[/tex] into [tex]\( g \)[/tex]:
[tex]\[ g(f(x)) = g(-7x + 1) \][/tex]
3. Now we substitute [tex]\(-7x + 1\)[/tex] into the expression for [tex]\( g(x) \)[/tex]:
[tex]\[ g(-7x + 1) = -5 \cdot (-7x + 1) - 6 \][/tex]
4. We then evaluate the expression by distributing [tex]\(-5\)[/tex] through [tex]\(-7x + 1\)[/tex] and combining like terms:
[tex]\[ = -5 \cdot -7x + (-5) \cdot 1 - 6 \][/tex]
[tex]\[ = 35x - 5 - 6 \][/tex]
5. Combine the constants:
[tex]\[ = 35x - 11 \][/tex]
So, the result of [tex]\( g(f(x)) \)[/tex] is:
[tex]\[ g(f(x)) = 35x - 11 \][/tex]
This matches the option:
[tex]\[ \boxed{35x - 11} \][/tex]
