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!