IDNLearn.com: Your one-stop destination for reliable answers to diverse questions. Get prompt and accurate answers to your questions from our community of experts who are always ready to help.

For the given functions [tex]\( f \)[/tex] and [tex]\( g \)[/tex], find the indicated composition:

[tex]\[ f(x) = -2x + 5, \quad g(x) = 4x + 7 \][/tex]

[tex]\[(g \circ f)(x)\][/tex]

A. [tex]\(-8x + 27\)[/tex]

B. [tex]\(8x + 27\)[/tex]

C. [tex]\(-8x + 19\)[/tex]

D. [tex]\(-8x - 13\)[/tex]


Sagot :

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]