To solve the problems, we need to evaluate the compositions of the functions [tex]\(f\)[/tex] and [tex]\(g\)[/tex] at the given points.
Given:
[tex]\[ f(x) = x^2 - 6x \][/tex]
[tex]\[ g(x) = x + 7 \][/tex]
Let's evaluate each part step-by-step.
### (a) [tex]\((f \circ g)(1)\)[/tex]
1. First, find [tex]\(g(1)\)[/tex]:
[tex]\[ g(1) = 1 + 7 = 8 \][/tex]
2. Next, use this result as the input to [tex]\(f\)[/tex]:
[tex]\[ f(g(1)) = f(8) \][/tex]
3. Now, evaluate [tex]\(f(8)\)[/tex]:
[tex]\[ f(8) = 8^2 - 6 \cdot 8 = 64 - 48 = 16 \][/tex]
Thus:
[tex]\[ (f \circ g)(1) = 16 \][/tex]
### (b) [tex]\((g \circ f)(1)\)[/tex]
1. First, find [tex]\(f(1)\)[/tex]:
[tex]\[ f(1) = 1^2 - 6 \cdot 1 = 1 - 6 = -5 \][/tex]
2. Next, use this result as the input to [tex]\(g\)[/tex]:
[tex]\[ g(f(1)) = g(-5) \][/tex]
3. Now, evaluate [tex]\(g(-5)\)[/tex]:
[tex]\[ g(-5) = -5 + 7 = 2 \][/tex]
Thus:
[tex]\[ (g \circ f)(1) = 2 \][/tex]
### Final Answer
So the evaluated results are:
(a) [tex]\((f \circ g)(1) = 16\)[/tex]
(b) [tex]\((g \circ f)(1) = 2\)[/tex]
