Question: Given the functions:
$f(x) = x^3 - 8x$
$g(x) = \sqrt{2x}$
$h(x) = 9x + 7$
Evaluate the function $g(f(x))$ for $x = 5$. Write your answer in exact simplified form. Select "Undefined" if applicable.
g(f(5)) is
1. **State the problem:** We need to find $g(f(5))$ given the functions $f(x) = x^3 - 8x$ and $g(x) = \sqrt{2x}$.
2. **Find $f(5)$:** Substitute $x = 5$ into $f(x)$:
$$f(5) = 5^3 - 8 \times 5 = 125 - 40 = 85$$
3. **Evaluate $g(f(5)) = g(85)$:** Substitute $85$ into $g(x)$:
$$g(85) = \sqrt{2 \times 85} = \sqrt{170}$$
4. **Simplify $\sqrt{170}$ if possible:**
Factor $170$:
$$170 = 2 \times 5 \times 17$$
None of these factors are perfect squares, so $\sqrt{170}$ is already in simplest radical form.
5. **Check domain:** Since $g(x) = \sqrt{2x}$, the expression inside the square root must be $\geq 0$.
Here, $2 \times 85 = 170 > 0$, so $g(85)$ is defined.
**Final answer:**
$$g(f(5)) = \sqrt{170}$$