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Let's first analyze the graph to identify the functions \( f \) and \( g \), and then solve the composition of functions listed.

- \( f(x) \): The blue linear function. It has a positive slope, and from the graph, we can estimate its equation. The line passes through points like \( (0, -2) \) and \( (2, 2) \), so its slope is \( \frac{2 - (-2)}{2 - 0} = 2 \). Thus, the equation of \( f(x) \) seems to be \( f(x) = 2x - 2 \).

- \( g(x) \): The red parabolic function. It appears to have a vertex at \( (2, -4) \) and opens upwards. The equation might resemble a form of \( g(x) = a(x - 2)^2 - 4 \). By estimating the value at other points, such as \( g(0) = 0 \), we can calculate that \( g(x) \approx (x - 2)^2 - 4 \).

Now, let's solve the problems:

1. \( (f \circ g)(2) \):
   - First, find \( g(2) \). From the graph, \( g(2) = -4 \).
   - Now, calculate \( f(g(2)) = f(-4) \). Using \( f(x) = 2x - 2 \), \( f(-4) = 2(-4) - 2 = -8 - 2 = -10 \).
   - Answer: \( -10 \).

2. \( (g \circ f)(2) \):
   - First, find \( f(2) \). Using \( f(x) = 2x - 2 \), \( f(2) = 2(2) - 2 = 4 - 2 = 2 \).
   - Now, calculate \( g(f(2)) = g(2) \). From the graph, \( g(2) = -4 \).
   - Answer: \( -4 \).

3. \( (f \circ f)(2) \):
   - First, find \( f(2) = 2 \) (from previous calculation).
   - Now, calculate \( f(f(2)) = f(2) \). From the previous step, we know \( f(2) = 2 \).
   - Answer: \( 2 \).

4. \( (g \circ g)(2) \):
   - First, find \( g(2) = -4 \) (from the graph).
   - Now, calculate \( g(g(2)) = g(-4) \). From the graph, estimate \( g(-4) \). It looks to be around \( 12 \) (as it's far above 10 on the graph).
   - Answer: \( 12 \).

5. \( (f + g)(4) \):
   - First, find \( f(4) \). Using \( f(x) = 2x - 2 \), \( f(4) = 2(4) - 2 = 8 - 2 = 6 \).
   - Now, find \( g(4) \). From the graph, \( g(4) = 0 \).
   - So, \( f(4) + g(4) = 6 + 0 = 6 \).
   - Answer: \( 6 \).

6. \( \left(\frac{f}{g}\right)(2) \):
   - We know \( f(2) = 2 \) and \( g(2) = -4 \).
   - So, \( \frac{f(2)}{g(2)} = \frac{2}{-4} = -\frac{1}{2} \).
   - Answer: \( -\frac{1}{2} \).

Would you like further details on any of these steps or explanations?

Here are 5 related questions you could explore:
1. What is the equation of a parabola given its vertex and one other point?
2. How do you calculate the composition of two functions in general terms?
3. How would the answers change if \( f(x) \) had a different slope?
4. Can you always estimate function values from a graph accurately?
5. How does the order of composition \( f \circ g \) differ from \( g \circ f \)?

**Tip:** Always check the graph carefully for critical points, such as intercepts and vertices, when identifying function values!

Evaluate the following function compositions and operations based on the given graph of a linear function f and a quadratic function g.

Learn how to evaluate the composition of a linear function f and a quadratic function g from a graph. The problem includes evaluating (f ∘ g)(2), (g ∘ f)(2), and other operations such as f + g and f / g. This is suitable for students in Grades 9-11.

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