Let's solve the given problems step by step.

Given functions:

• $f(x) = 2x - 4$
• $g(x) = 6 - x$

We need to evaluate these functions at specified points or perform operations involving these functions.

(a) $(f + g)(2)$

To find $(f + g)(2)$, we add $f(2)$ and $g(2)$:

1. Calculate $f(2)$: $f(2) = 2(2) - 4 = 4 - 4 = 0$

2. Calculate $g(2)$: $g(2) = 6 - 2 = 4$

3. Therefore, $(f + g)(2) = f(2) + g(2) = 0 + 4 = 4$

(b) $(f - g)(-6)$

To find $(f - g)(-6)$, we subtract $g(-6)$ from $f(-6)$:

1. Calculate $f(-6)$: $f(-6) = 2(-6) - 4 = -12 - 4 = -16$

2. Calculate $g(-6)$: $g(-6) = 6 - (-6) = 6 + 6 = 12$

3. Therefore, $(f - g)(-6) = f(-6) - g(-6) = -16 - 12 = -28$

(c) $(g - f)(1)$

To find $(g - f)(1)$, we subtract $f(1)$ from $g(1)$:

1. Calculate $g(1)$: $g(1) = 6 - 1 = 5$

2. Calculate $f(1)$: $f(1) = 2(1) - 4 = 2 - 4 = -2$

3. Therefore, $(g - f)(1) = g(1) - f(1) = 5 - (-2) = 5 + 2 = 7$

(d) $(g \cdot f)\left(\frac{1}{2}\right)$

To find $(g \cdot f)\left(\frac{1}{2}\right)$, we multiply $g\left(\frac{1}{2}\right)$ by $f\left(\frac{1}{2}\right)$:

1. Calculate $g\left(\frac{1}{2}\right)$: $g\left(\frac{1}{2}\right) = 6 - \frac{1}{2} = \frac{12}{2} - \frac{1}{2} = \frac{11}{2}$

2. Calculate $f\left(\frac{1}{2}\right)$: $f\left(\frac{1}{2}\right) = 2\left(\frac{1}{2}\right) - 4 = 1 - 4 = -3$

3. Therefore, $(g \cdot f)\left(\frac{1}{2}\right) = g\left(\frac{1}{2}\right) \cdot f\left(\frac{1}{2}\right) = \frac{11}{2} \cdot (-3) = -\frac{33}{2}$

(e) $\left(\frac{f}{g}\right)(-4)$

To find $\left(\frac{f}{g}\right)(-4)$, we divide $f(-4)$ by $g(-4)$:

1. Calculate $f(-4)$: $f(-4) = 2(-4) - 4 = -8 - 4 = -12$

2. Calculate $g(-4)$: $g(-4) = 6 - (-4) = 6 + 4 = 10$

3. Therefore, $\left(\frac{f}{g}\right)(-4) = \frac{f(-4)}{g(-4)} = \frac{-12}{10} = -\frac{6}{5}$

(f) $\left(\frac{g}{f}\right)(-3)$

To find $\left(\frac{g}{f}\right)(-3)$, we divide $g(-3)$ by $f(-3)$:

1. Calculate $g(-3)$: $g(-3) = 6 - (-3) = 6 + 3 = 9$

2. Calculate $f(-3)$: $f(-3) = 2(-3) - 4 = -6 - 4 = -10$

3. Therefore, $\left(\frac{g}{f}\right)(-3) = \frac{g(-3)}{f(-3)} = \frac{9}{-10} = -\frac{9}{10}$

Final Results:

• (a) $(f + g)(2) = 4$
• (b) $(f - g)(-6) = -28$
• (c) $(g - f)(1) = 7$
• (d) $(g \cdot f)\left(\frac{1}{2}\right) = -\frac{33}{2}$
• (e) $\left(\frac{f}{g}\right)(-4) = -\frac{6}{5}$
• (f) $\left(\frac{g}{f}\right)(-3) = -\frac{9}{10}$

Would you like further details or have any questions?

Follow-up Questions:

1. What is the process for finding the sum of two functions?
2. How do you evaluate a function at a specific point?
3. What does it mean to find the difference between two functions?
4. How is multiplication between two functions performed?
5. What steps are involved in dividing one function by another?

Tip:

Always carefully substitute values into functions and simplify step-by-step to avoid calculation errors.