Given the functions: f(x)=5x g(x)=|x+7| h(x)=1/x-2 Evaluate the function (h/g)(5) for x=5. Write your answer as an integer or simplified fraction. Select "Undefined" if applicable. (h/g)(5) is

We are tasked with evaluating \((h/g)(5)\), which represents the quotient of the functions \(h(x)\) and \(g(x)\) evaluated at \(x = 5\). The functions are:

- \( f(x) = 5x \) (not needed for this particular calculation)
- \( g(x) = |x + 7| \)
- \( h(x) = \frac{1}{x} - 2 \)

### Step 1: Evaluate \( g(5) \)
We first evaluate \( g(x) \) at \( x = 5 \):
\[
g(5) = |5 + 7| = |12| = 12
\]

### Step 2: Evaluate \( h(5) \)
Next, we evaluate \( h(x) \) at \( x = 5 \):
\[
h(5) = \frac{1}{5} - 2 = \frac{1}{5} - \frac{10}{5} = \frac{-9}{5}
\]

### Step 3: Calculate \((h/g)(5)\)
Now, we calculate the quotient \((h/g)(5)\):
\[
(h/g)(5) = \frac{h(5)}{g(5)} = \frac{\frac{-9}{5}}{12} = \frac{-9}{5} \times \frac{1}{12} = \frac{-9}{60} = \frac{-3}{20}
\]

Thus, \((h/g)(5) = \frac{-3}{20}\).

The final answer is:
\[
\boxed{\frac{-3}{20}}
\]

Would you like further clarification or details on any of the steps? Here are five related questions to extend the understanding:

1. How would the result change if \(h(x)\) were \(1/x + 2\) instead of \(1/x - 2\)?
2. What would \((g/h)(5)\) be with the same functions?
3. How does the absolute value in \(g(x)\) affect the evaluation at negative values?
4. What happens to \(h(x)\) as \(x\) approaches zero?
5. Can you evaluate \((h/g)(-7)\) and explain why it might be undefined?

**Tip**: Always simplify the quotient of functions step-by-step, starting with each function's individual value at the given point.

Given the functions: f(x)=5x, g(x)=|x+7|, h(x)=1/x-2, Evaluate the function (h/g)(5) for x=5. Write your answer as an integer or simplified fraction. Select 'Undefined' if applicable.

Learn how to evaluate the quotient of functions (h/g)(5) given the functions f(x)=5x, g(x)=|x+7|, and h(x)=1/x-2. This problem focuses on algebraic concepts like absolute value and rational functions. Step-by-step solution provided for students in Grades 9-11.

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