To find [tex]\((f \circ g)(x)\)[/tex], we need to evaluate the composition of the functions [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex]. In other words, we need to find [tex]\(f(g(x))\)[/tex], which means we will substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex].
First, let's write down the functions:
[tex]\[ f(x) = \frac{x-1}{3} \][/tex]
[tex]\[ g(x) = 3x + 1 \][/tex]
Now, we will substitute [tex]\(g(x)\)[/tex] into [tex]\(f(x)\)[/tex]:
[tex]\[ f(g(x)) = f(3x + 1) \][/tex]
Next, we replace [tex]\(x\)[/tex] in [tex]\(f(x)\)[/tex] with [tex]\(g(x)\)[/tex], which in this case is [tex]\(3x + 1\)[/tex]:
[tex]\[ f(3x + 1) = \frac{(3x + 1) - 1}{3} \][/tex]
Simplify within the parentheses:
[tex]\[ f(3x + 1) = \frac{3x + 1 - 1}{3} \][/tex]
[tex]\[ f(3x + 1) = \frac{3x}{3} \][/tex]
Finally, simplify the fraction:
[tex]\[ f(3x + 1) = x \][/tex]
Thus, the function [tex]\((f \circ g)(x) = f(g(x)) = x\)[/tex].
So, the correct answer is:
[tex]\[ (f \circ g)(x) = x \][/tex]
