Sure, let's tackle each part of the question step-by-step.
### Definitions:
Given the functions:
- [tex]\( f(x) = 6 - 2x \)[/tex]
- [tex]\( g(x) = \frac{9}{x} \)[/tex]
- [tex]\( h(x) = 6 + x^2 \)[/tex]
### a) Evaluate [tex]\( f(-1) \)[/tex]:
To find [tex]\( f(-1) \)[/tex]:
[tex]\[
f(-1) = 6 - 2(-1) = 6 + 2 = 8
\][/tex]
### b) Evaluate [tex]\( g\left(\frac{1}{3}\right) \)[/tex]:
To find [tex]\( g\left(\frac{1}{3}\right) \)[/tex]:
[tex]\[
g\left(\frac{1}{3}\right) = \frac{9}{\frac{1}{3}} = 9 \times 3 = 27
\][/tex]
### c) [tex]\( h(f(x)) \)[/tex]:
To express [tex]\( h(f(x)) \)[/tex], we first find [tex]\( f(x) \)[/tex] and then substitute it into [tex]\( h(x) \)[/tex].
[tex]\[
f(x) = 6 - 2x
\][/tex]
Now substitute [tex]\( f(x) \)[/tex] into [tex]\( h(x) \)[/tex]:
[tex]\[
h(f(x)) = h(6 - 2x) = 6 + (6 - 2x)^2
\][/tex]
Simplify the expression inside the square:
[tex]\[
(6 - 2x)^2 = 36 - 24x + 4x^2
\][/tex]
Thus,
[tex]\[
h(6 - 2x) = 6 + 36 - 24x + 4x^2 = 42 - 24x + 4x^2
\][/tex]
So,
[tex]\[
h(f(x)) = 42 - 24x + 4x^2
\][/tex]
### d) [tex]\( gf(x) \)[/tex]:
To express [tex]\( gf(x) \)[/tex], we find [tex]\( f(x) \)[/tex] first and then substitute it into [tex]\( g(x) \)[/tex]:
[tex]\[
f(x) = 6 - 2x
\][/tex]
Now substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(f(x)) = g(6 - 2x) = \frac{9}{6 - 2x}
\][/tex]
So,
[tex]\[
g(f(x)) = \frac{9}{6 - 2x}
\][/tex]
### e) Evaluate [tex]\( f(g(12)) \)[/tex]:
To find [tex]\( f(g(12)) \)[/tex]:
First, evaluate [tex]\( g(12) \)[/tex]:
[tex]\[
g(12) = \frac{9}{12} = \frac{3}{4}
\][/tex]
Now, substitute this value into [tex]\( f(x) \)[/tex]:
[tex]\[
f\left(\frac{3}{4}\right) = 6 - 2 \left(\frac{3}{4}\right) = 6 - \frac{3}{2} = 6 - 1.5 = 4.5
\][/tex]
### f) Evaluate [tex]\( h(f(-9)) \)[/tex]:
To find [tex]\( h(f(-9)) \)[/tex]:
First, evaluate [tex]\( f(-9) \)[/tex]:
[tex]\[
f(-9) = 6 - 2(-9) = 6 + 18 = 24
\][/tex]
Now, substitute this value into [tex]\( h(x) \)[/tex]:
[tex]\[
h(24) = 6 + 24^2 = 6 + 576 = 582
\][/tex]
### Summary of Results:
[tex]\[
\begin{aligned}
&\text{a) } f(-1) = 8 \\
&\text{b) } g\left(\frac{1}{3}\right) = 27 \\
&\text{c) } h(f(x)) = 42 - 24x + 4x^2 \\
&\text{d) } g(f(x)) = \frac{9}{6 - 2x} \\
&\text{e) } f(g(12)) = 4.5 \\
&\text{f) } h(f(-9)) = 582 \\
\end{aligned}
\][/tex]
