https://file.aiquickdraw.com/mathbot/file/17259885893238siojk1d.jpg

The problem in the image involves function composition. We are given two functions:

\[
F(x) = \frac{2x}{x + 3} \quad \text{and} \quad G(x) = x - 1
\]

We are tasked with finding \( (F \circ G)(x) \), which is the composition of \( F \) and \( G \), i.e., \( F(G(x)) \).

### Step-by-step solution:

1. **Substitute \( G(x) = x - 1 \) into \( F(x) \):**

\[
F(G(x)) = F(x - 1)
\]

2. **Use the definition of \( F(x) = \frac{2x}{x + 3} \):**

Now, substitute \( x - 1 \) wherever there is \( x \) in \( F(x) \):

\[
F(x - 1) = \frac{2(x - 1)}{(x - 1) + 3}
\]

3. **Simplify the expression:**

\[
F(x - 1) = \frac{2(x - 1)}{x - 1 + 3} = \frac{2(x - 1)}{x + 2}
\]

Expand the numerator:

\[
F(x - 1) = \frac{2x - 2}{x + 2}
\]

This is the formula for \( (F \circ G)(x) \).

### Answer:

The correct formula for \( (F \circ G)(x) \) is:

\[
\boxed{c) \frac{2x - 2}{x + 2}}
\]

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Would you like any additional details or explanations?

Here are 5 related questions you could explore:
1. How do you evaluate the composition of two functions in general?
2. What happens if \( G(x) = x + 2 \) instead of \( x - 1 \)?
3. Can the composition of functions be reversed, i.e., \( (G \circ F)(x) \)?
4. How does function composition differ from simple multiplication of functions?
5. Can \( F(x) \) or \( G(x) \) ever lead to an undefined expression for certain values of \( x \)?

**Tip:** When composing functions, always substitute carefully and simplify the resulting expressions step by step!

If F(x) = 2x / (x + 3) and G(x) = x - 1, find a formula for (F o G)(x).

Learn how to compose two functions: F(x) = 2x / (x + 3) and G(x) = x - 1. This algebra problem involves substituting G(x) into F(x) and simplifying the result, making it suitable for students in Grades 10-12.

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