To find the compositions [tex]\( (f \circ f)(x) \)[/tex] and [tex]\( (g \circ g)(x) \)[/tex], we need to substitute the functions into themselves. Let's go through each step-by-step.
### Composition [tex]\( (f \circ f)(x) \)[/tex]
Recall that the composition [tex]\( (f \circ f)(x) \)[/tex] means [tex]\( f(f(x)) \)[/tex].
1. Start with [tex]\( f(x) = \frac{8}{x} \)[/tex].
2. Now compute [tex]\( f(f(x)) \)[/tex]:
[tex]\[
f(f(x)) = f\left(\frac{8}{x}\right)
\][/tex]
3. Substitute [tex]\( \frac{8}{x} \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f\left(\frac{8}{x}\right) = \frac{8}{\frac{8}{x}}
\][/tex]
4. Simplify the expression:
[tex]\[
\frac{8}{\frac{8}{x}} = \frac{8 \cdot x}{8} = x
\][/tex]
Therefore, the composition [tex]\( (f \circ f)(x) = x \)[/tex].
### Composition [tex]\( (g \circ g)(x) \)[/tex]
Now, let’s consider the composition [tex]\( (g \circ g)(x) \)[/tex] which means [tex]\( g(g(x)) \)[/tex].
1. Start with [tex]\( g(x) = x^2 - 3 \)[/tex].
2. Compute [tex]\( g(g(x)) \)[/tex]:
[tex]\[
g(g(x)) = g(x^2 - 3)
\][/tex]
3. Substitute [tex]\( x^2 - 3 \)[/tex] into [tex]\( g(x) \)[/tex]:
[tex]\[
g(x^2 - 3) = (x^2 - 3)^2 - 3
\][/tex]
4. Expand [tex]\((x^2 - 3)^2 \)[/tex]:
[tex]\[
(x^2 - 3)^2 = (x^2 - 3) \cdot (x^2 - 3)
\][/tex]
[tex]\[
= x^4 - 3x^2 - 3x^2 + 9
\][/tex]
[tex]\[
= x^4 - 6x^2 + 9
\][/tex]
5. Now subtract 3 to simplify the final expression:
[tex]\[
(x^4 - 6x^2 + 9) - 3 = x^4 - 6x^2 + 6
\][/tex]
Therefore, the composition [tex]\( (g \circ g)(x) = x^4 - 6x^2 + 6 \)[/tex].
### Final Answers
[tex]\[
(f \circ f)(x) = x
\][/tex]
[tex]\[
(g \circ g)(x) = x^4 - 6x^2 + 6
\][/tex]
