To find the compositions [tex]\((h \circ h)(x)\)[/tex] and [tex]\((f \circ f)(x)\)[/tex] given the functions [tex]\( h(x) \)[/tex] and [tex]\( f(x) \)[/tex], let's proceed step by step.
### Composition [tex]\( (h \circ h)(x) \)[/tex]
1. Define [tex]\( h(x) \)[/tex]:
[tex]\[
h(x) = x^2 - 8
\][/tex]
2. Compute [tex]\( h(h(x)) \)[/tex] by substituting [tex]\( h(x) \)[/tex] into itself:
[tex]\[
(h \circ h)(x) = h(h(x))
\][/tex]
Substituting [tex]\( h(x) \)[/tex]:
[tex]\[
h(h(x)) = h(x^2 - 8)
\][/tex]
3. Evaluate [tex]\( h(x^2 - 8) \)[/tex] using the definition of [tex]\( h(x) \)[/tex]:
[tex]\[
h(x^2 - 8) = (x^2 - 8)^2 - 8
\][/tex]
4. Expand the expression [tex]\((x^2 - 8)^2\)[/tex]:
[tex]\[
(x^2 - 8)^2 = x^4 - 16x^2 + 64
\][/tex]
5. Subtract 8 from the result:
[tex]\[
x^4 - 16x^2 + 64 - 8
\][/tex]
6. Simplify the expression:
[tex]\[
x^4 - 16x^2 + 56
\][/tex]
Thus:
[tex]\[
(h \circ h)(x) = x^4 - 16x^2 + 56
\][/tex]
### Composition [tex]\( (f \circ f)(x) \)[/tex]
1. Define [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = \frac{x}{7}
\][/tex]
2. Compute [tex]\( f(f(x)) \)[/tex] by substituting [tex]\( f(x) \)[/tex] into itself:
[tex]\[
(f \circ f)(x) = f(f(x))
\][/tex]
Substituting [tex]\( f(x) \)[/tex]:
[tex]\[
f(f(x)) = f\left(\frac{x}{7}\right)
\][/tex]
3. Evaluate [tex]\( f\left(\frac{x}{7}\right) \)[/tex] using the definition of [tex]\( f(x) \)[/tex]:
[tex]\[
f\left(\frac{x}{7}\right) = \frac{1}{7} \cdot \frac{x}{7}
\][/tex]
4. Simplify the expression:
[tex]\[
\frac{x}{7} \cdot \frac{1}{7} = \frac{x}{49}
\][/tex]
Thus:
[tex]\[
(f \circ f)(x) = \frac{x}{49}
\][/tex]
Summarizing the results:
[tex]\[
(h \circ h)(x) = x^4 - 16x^2 + 56
\][/tex]
[tex]\[
(f \circ f)(x) = \frac{x}{49}
\][/tex]
