Let's work through the problem step by step, using the graph of the functions $f(x)$ (the blue curve) and $g(x)$ (the red line) to answer the parts (a) through (f). I'll extract the values of $f(x)$ and $g(x)$ from the graph at the required points.

Values from the graph:

• $f(2) = 4$, $g(2) = 2$
• $f(4) = 0$, $g(4) = 4$
• $f(6) = 2$, $g(6) = 6$

Now, let's solve each part:

(a) $(f + g)(2)$ $(f + g)(2) = f(2) + g(2) = 4 + 2 = 6$

(b) $(f + g)(4)$ $(f + g)(4) = f(4) + g(4) = 0 + 4 = 4$

(c) $(f - g)(6)$ $(f - g)(6) = f(6) - g(6) = 2 - 6 = -4$

(d) $(g - f)(6)$ $(g - f)(6) = g(6) - f(6) = 6 - 2 = 4$

(e) $(f \cdot g)(2)$ $(f \cdot g)(2) = f(2) \cdot g(2) = 4 \cdot 2 = 8$

(f) $\left( \frac{f}{g} \right)(4)$ $\left( \frac{f}{g} \right)(4) = \frac{f(4)}{g(4)} = \frac{0}{4} = 0$

Answers:

• (a) 6
• (b) 4
• (c) -4
• (d) 4
• (e) 8
• (f) 0

Would you like any additional details or explanations?

Here are 5 related questions to deepen your understanding:

1. How would the values change if the graph were shifted vertically?
2. What does the product $(f \cdot g)(x)$ represent graphically?
3. How would you calculate $f(x) + g(x)$ at any arbitrary point?
4. How does the graph of $f(x)$ differ from $g(x)$ in terms of its general behavior?
5. Could there be points where $f(x) = g(x)$ based on the graph?

Tip: Always check the graph carefully to ensure correct values at specific points—this ensures the accuracy of your calculations!