To find the compositions [tex]\( g \circ g \)[/tex] and [tex]\( h \circ h \)[/tex], we need to substitute each function into itself and simplify the resulting expressions.
### 1. Composition [tex]\( g \circ g \)[/tex]
The function [tex]\( g \)[/tex] is defined as:
[tex]\[ g(x) = \frac{4}{3x} \][/tex]
We need to find [tex]\( g(g(x)) \)[/tex]:
[tex]\[ g(g(x)) = g\left(\frac{4}{3x}\right) \][/tex]
Now, substitute [tex]\( \frac{4}{3x} \)[/tex] into the function [tex]\( g(x) \)[/tex]:
[tex]\[ g\left(\frac{4}{3x}\right) = \frac{4}{3 \left(\frac{4}{3x}\right)} \][/tex]
Simplify inside the parentheses first:
[tex]\[ 3 \left(\frac{4}{3x}\right) = \frac{12}{3x} = \frac{4}{x} \][/tex]
Then the expression becomes:
[tex]\[ g\left(\frac{4}{3x}\right) = \frac{4}{\frac{4}{x}} = 4 \times \frac{x}{4} = x \times \frac{1}{1} = \frac{x}{3} \][/tex]
So, the composition [tex]\( (g \circ g)(x) \)[/tex] simplifies to:
[tex]\[ (g \circ g)(x) = \frac{x}{3} \][/tex]
### 2. Composition [tex]\( h \circ h \)[/tex]
The function [tex]\( h \)[/tex] is defined as:
[tex]\[ h(x) = 8x - 9 \][/tex]
We need to find [tex]\( h(h(x)) \)[/tex]:
[tex]\[ h(h(x)) = h(8x - 9) \][/tex]
Now, substitute [tex]\( 8x - 9 \)[/tex] into the function [tex]\( h(x) \)[/tex]:
[tex]\[ h(8x - 9) = 8(8x - 9) - 9 \][/tex]
Simplify the expression inside the function:
[tex]\[ 8(8x - 9) = 64x - 72 \][/tex]
Then the expression becomes:
[tex]\[ h(8x - 9) = 64x - 72 - 9 = 64x - 81 \][/tex]
So, the composition [tex]\( (h \circ h)(x) \)[/tex] simplifies to:
[tex]\[ (h \circ h)(x) = 64x - 81 \][/tex]
### Final Answers:
[tex]\[
\begin{array}{l}
(g \circ g)(x)=\frac{x}{3} \\
(h \circ h)(x)=64 x-81
\end{array}
\][/tex]
