Sure, let's work through the problem step-by-step to find the composition [tex]\((g \circ f)(x)\)[/tex].
We are given the functions:
[tex]\[ f(x) = -2x + 5 \][/tex]
[tex]\[ g(x) = 4x + 7 \][/tex]
The composition [tex]\((g \circ f)(x)\)[/tex] means we need to evaluate [tex]\(g(f(x))\)[/tex].
1. Evaluate [tex]\(f(x)\)[/tex]:
[tex]\[ f(x) = -2x + 5 \][/tex]
2. Substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
Next, we substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex], that is, we use the expression we found for [tex]\(f(x)\)[/tex] as the input to [tex]\(g(x)\)[/tex].
[tex]\[ g(f(x)) = g(-2x + 5) \][/tex]
3. Compute [tex]\(g(-2x + 5)\)[/tex]:
We now substitute [tex]\(-2x + 5\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[ g(-2x + 5) = 4(-2x + 5) + 7 \][/tex]
4. Simplify the expression:
We distribute and combine like terms:
[tex]\[ g(-2x + 5) = 4(-2x) + 4(5) + 7 \][/tex]
[tex]\[ g(-2x + 5) = -8x + 20 + 7 \][/tex]
[tex]\[ g(-2x + 5) = -8x + 27 \][/tex]
Therefore, the resulting expression for the composition [tex]\((g \circ f)(x)\)[/tex] is:
[tex]\[ (g \circ f)(x) = -8x + 27 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{-8x + 27} \][/tex]
Therefore, the correct option is:
A. [tex]\(-8x + 27\)[/tex]
