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Part 1
Let f​(x​)equals x squared plus 8 x and g​(x​)equals 7 minus x. 
Find left parenthesis StartFraction f Over g EndFraction right parenthesis left parenthesis x right parenthesis
 and left parenthesis StartFraction f Over g EndFraction right parenthesis
​(4​).
Question content area bottom
Part 1
left parenthesis StartFraction f Over g EndFraction right parenthesis left parenthesis x right parenthesis
equals
  
enter your response here ​(Simplify your​ answer.)

To solve this, we are asked to find the expression for \( \left( \frac{f}{g} \right)(x) \) and also to evaluate \( \left( \frac{f}{g} \right)(4) \).

### Step 1: Define the functions
We are given two functions:

- \( f(x) = x^2 + 8x \)
- \( g(x) = 7 - x \)

### Step 2: Form the ratio \( \frac{f(x)}{g(x)} \)
To find \( \left( \frac{f}{g} \right)(x) \), we simply divide \( f(x) \) by \( g(x) \):
\[
\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{x^2 + 8x}{7 - x}
\]

### Step 3: Evaluate \( \left( \frac{f}{g} \right)(4) \)
Now, we need to substitute \( x = 4 \) into the expression \( \frac{x^2 + 8x}{7 - x} \):

1. Calculate \( f(4) \):
\[
f(4) = 4^2 + 8(4) = 16 + 32 = 48
\]

2. Calculate \( g(4) \):
\[
g(4) = 7 - 4 = 3
\]

3. Divide \( f(4) \) by \( g(4) \):
\[
\left( \frac{f}{g} \right)(4) = \frac{48}{3} = 16
\]

### Final Answer:
- \( \left( \frac{f}{g} \right)(x) = \frac{x^2 + 8x}{7 - x} \)
- \( \left( \frac{f}{g} \right)(4) = 16 \)

Would you like further details on this, or do you have any questions?

Here are 5 related questions for practice:
1. Find \( \left( \frac{f}{g} \right)(-2) \).
2. What happens to \( \left( \frac{f}{g} \right)(x) \) as \( x \to 7 \)?
3. Simplify \( \frac{x^2 + 8x}{7 - x} \) for large values of \( x \).
4. What is the domain of \( \left( \frac{f}{g} \right)(x) \)?
5. Find \( f(0) \) and \( g(0) \), then calculate \( \left( \frac{f}{g} \right)(0) \).

**Tip**: When simplifying expressions involving fractions, always check for values of \( x \) that make the denominator zero, as these are points where the function is undefined.

Let f(x) = x^2 + 8x and g(x) = 7 - x. Find (f/g)(x) and (f/g)(4).

Learn how to find and simplify the expression (f/g)(x) where f(x) = x^2 + 8x and g(x) = 7 - x. This problem involves dividing two functions and evaluating the result at x = 4. Suitable for students in Grades 9-10.

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