Let's solve the given problems for each function: $f(x)$, $g(x)$, and $h(x)$.

Given Functions:

1. $f(x) = | -14 + 8x |$
2. $g(x) = \sqrt{x - 2} + 3$
3. $h(x) = \frac{3 + x^2}{x + 7}$

Problems:

We need to find $f\left(\frac{1}{4}\right)$, $g(5)$, and $h(5)$.

Solutions:

1. Finding $f\left(\frac{1}{4}\right)$:

Substitute $x = \frac{1}{4}$ into $f(x)$: $f\left(\frac{1}{4}\right) = \left| -14 + 8 \cdot \frac{1}{4} \right|$ Simplify inside the absolute value: $= \left| -14 + 2 \right| = \left| -12 \right| = 12$ So, $f\left(\frac{1}{4}\right) = 12$.

2. Finding $g(5)$:

Substitute $x = 5$ into $g(x)$: $g(5) = \sqrt{5 - 2} + 3$ Simplify inside the square root: $= \sqrt{3} + 3$ So, $g(5) = \sqrt{3} + 3$.

3. Finding $h(5)$:

Substitute $x = 5$ into $h(x)$: $h(5) = \frac{3 + 5^2}{5 + 7}$ Simplify: $= \frac{3 + 25}{12} = \frac{28}{12} = \frac{7}{3}$ So, $h(5) = \frac{7}{3}$.

Final Answers:

• $f\left(\frac{1}{4}\right) = 12$
• $g(5) = \sqrt{3} + 3$
• $h(5) = \frac{7}{3}$

Would you like a more detailed explanation on any of these steps?

Here are some related questions to explore further:

1. How does the absolute value function affect the shape of $f(x)$?
2. What are the domain and range of $g(x) = \sqrt{x - 2} + 3$?
3. How would $h(x)$ behave as $x \to -7$?
4. Can $f(x)$ have any negative outputs?
5. How does the square root in $g(x)$ affect the rate of increase compared to a linear function?

Tip: When working with functions involving absolute values or square roots, always consider restrictions on the domain to ensure valid inputs.