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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?

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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!

Given the functions f(x) = -2x, g(x) = |x + 7|, and h(x) = 1 / (x - 5), evaluate the function (h / f)(-5).

Learn how to evaluate (h / f)(-5) by dividing two functions, f(x) = -2x and h(x) = 1 / (x - 5), using function substitution and simplification. This problem explores concepts of function division, substitution, and evaluation, suitable for Grades 9-12.

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