We are given three functions:

• $f(x) = -2x$
• $g(x) = |x + 7|$
• $h(x) = \frac{1}{x - 5}$

We are asked to evaluate $\left( \frac{h}{f} \right)(-5)$, meaning we need to compute $\frac{h(-5)}{f(-5)}$.

Step 1: Evaluate $h(-5)$

The function $h(x) = \frac{1}{x - 5}$.

For $x = -5$: $h(-5) = \frac{1}{-5 - 5} = \frac{1}{-10} = -\frac{1}{10}$

Step 2: Evaluate $f(-5)$

The function $f(x) = -2x$.

For $x = -5$: $f(-5) = -2(-5) = 10$

Step 3: Compute $\frac{h(-5)}{f(-5)}$

$\frac{h(-5)}{f(-5)} = \frac{-\frac{1}{10}}{10} = -\frac{1}{10} \times \frac{1}{10} = -\frac{1}{100}$

Thus, the value of $\left( \frac{h}{f} \right)(-5)$ is $-\frac{1}{100}$.

Do you need further clarification or additional details?

Here are five related questions for practice:

1. How would you evaluate $\left( \frac{g}{f} \right)(-5)$?
2. What is the domain of the function $h(x) = \frac{1}{x - 5}$?
3. How do you compute $\left( \frac{h}{g} \right)(-5)$?
4. What is the value of $g(-5)$ for $g(x) = |x + 7|$?
5. How would you simplify the expression $\frac{1}{10} \times \frac{1}{10}$?

Tip:

When dividing fractions, remember to multiply by the reciprocal of the denominator!