https://file.aiquickdraw.com/mathbot/file/172759498310189an33jd.jpg

The problem in the image asks us to evaluate the following function expression:

\[
f(x) = \frac{x^2 + 4x}{x + 6}
\]

Then, we are required to find:

\[
\frac{f(0) + f(2)}{f(2) + f(3)}
\]

### Step 1: Find \( f(0) \)
Substitute \( x = 0 \) into the function:

\[
f(0) = \frac{0^2 + 4(0)}{0 + 6} = \frac{0}{6} = 0
\]

### Step 2: Find \( f(2) \)
Substitute \( x = 2 \) into the function:

\[
f(2) = \frac{2^2 + 4(2)}{2 + 6} = \frac{4 + 8}{8} = \frac{12}{8} = \frac{3}{2}
\]

### Step 3: Find \( f(3) \)
Substitute \( x = 3 \) into the function:

\[
f(3) = \frac{3^2 + 4(3)}{3 + 6} = \frac{9 + 12}{9} = \frac{21}{9} = \frac{7}{3}
\]

### Step 4: Evaluate the expression \( \frac{f(0) + f(2)}{f(2) + f(3)} \)

Now, substitute the values we found for \( f(0) \), \( f(2) \), and \( f(3) \):

\[
\frac{f(0) + f(2)}{f(2) + f(3)} = \frac{0 + \frac{3}{2}}{\frac{3}{2} + \frac{7}{3}}
\]

To simplify, let's first add \( \frac{3}{2} \) and \( \frac{7}{3} \) in the denominator. The common denominator is 6:

\[
\frac{3}{2} + \frac{7}{3} = \frac{9}{6} + \frac{14}{6} = \frac{23}{6}
\]

Thus, the full expression becomes:

\[
\frac{\frac{3}{2}}{\frac{23}{6}} = \frac{3}{2} \times \frac{6}{23} = \frac{18}{46} = \frac{9}{23}
\]

### Final Answer:

\[
\frac{f(0) + f(2)}{f(2) + f(3)} = \frac{9}{23}
\]

Let me know if you have any questions or need further details!

### Related Questions:
1. How would the result change if the denominator function were different?
2. What is the domain of the function \( f(x) = \frac{x^2 + 4x}{x + 6} \)?
3. How can you simplify more complex rational functions like this one?
4. What happens to \( f(x) \) as \( x \to -6 \)?
5. How do you solve function expressions involving complex fractions?

### Tip:
Always simplify fractions as much as possible before substituting values, as this can make the calculations easier!

If f(x) = (x^2 + 4x) / (x + 6), find (f(0) + f(2)) / (f(2) + f(3))

Learn how to evaluate the function f(x) = (x^2 + 4x) / (x + 6) for specific values of x, and solve the expression (f(0) + f(2)) / (f(2) + f(3)). This problem demonstrates the use of substitution and simplification of rational functions, suitable for Grades 9-11.

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