To find [tex]\((g \circ f)(x)\)[/tex], we need to substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex] and then simplify the resulting expression. Let’s go through the steps in detail:
1. Define [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]:
[tex]\[
f(x) = \frac{x-5}{4}
\][/tex]
[tex]\[
g(x) = 4x + 5
\][/tex]
2. Substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
We need to find [tex]\(g(f(x))\)[/tex], which means we substitute [tex]\(f(x)\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[
g(f(x)) = g\left(\frac{x-5}{4}\right)
\][/tex]
3. Evaluate [tex]\(g(f(x))\)[/tex]:
Substitute [tex]\(\frac{x-5}{4}\)[/tex] into [tex]\(g(x)\)[/tex]:
[tex]\[
g\left(\frac{x-5}{4}\right) = 4 \left(\frac{x-5}{4}\right) + 5
\][/tex]
4. Simplify the expression:
[tex]\[
4 \left(\frac{x-5}{4}\right) + 5 = x - 5 + 5
\][/tex]
[tex]\[
x - 5 + 5 = x
\][/tex]
5. Result:
Therefore, [tex]\((g \circ f)(x) = x\)[/tex].
Thus, the correct answer is:
[tex]\(
\boxed{x}
\)[/tex]
