To find the value of [tex]\(\left(\frac{f}{g}\right)(8)\)[/tex] where [tex]\(f(x) = 3 - 2x\)[/tex] and [tex]\(g(x) = \frac{1}{x+5}\)[/tex], we need to follow these steps:
1. Evaluate [tex]\(f(8)\)[/tex]:
[tex]\[ f(x) = 3 - 2x \][/tex]
Substitute [tex]\(x = 8\)[/tex]:
[tex]\[ f(8) = 3 - 2 \cdot 8 \][/tex]
[tex]\[ f(8) = 3 - 16 \][/tex]
[tex]\[ f(8) = -13 \][/tex]
2. Evaluate [tex]\(g(8)\)[/tex]:
[tex]\[ g(x) = \frac{1}{x+5} \][/tex]
Substitute [tex]\(x = 8\)[/tex]:
[tex]\[ g(8) = \frac{1}{8 + 5} \][/tex]
[tex]\[ g(8) = \frac{1}{13} \][/tex]
3. Compute the value of [tex]\(\left(\frac{f}{g}\right)(8)\)[/tex]:
[tex]\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}
\][/tex]
Substitute [tex]\(x = 8\)[/tex]:
[tex]\[
\left(\frac{f}{g}\right)(8) = \frac{f(8)}{g(8)} = \frac{-13}{\frac{1}{13}}
\][/tex]
4. Simplify the expression:
[tex]\[
\frac{-13}{\frac{1}{13}} = -13 \times 13 = -169
\][/tex]
So, the value of [tex]\(\left(\frac{f}{g}\right)(8)\)[/tex] is [tex]\(\boxed{-169}\)[/tex].
