Questions: Let f(x)=2x-6 and g(x)=8-x. Find the following. (a) (f+g)(x) (b) (f-g)(x) (c) (f⋅g)(x) (d) (f/g) (e) The domain of (f/g) (a) (f+g)(x)= (Simplify your answer.) (b) (f-g)(x)= (Simplify your answer.) (c) (f⋅g)(x)= (Simplify your answer.) (d) ((f/g))(x)= (Simplify your answer.) (e) The domain of (f/g) is . (Type your answer in interval notation.)

Let f(x)=2x-6 and g(x)=8-x. Find the following.
(a) (f+g)(x)
(b) (f-g)(x)
(c) (f⋅g)(x)
(d) (f/g)
(e) The domain of (f/g)
(a) (f+g)(x)= (Simplify your answer.)
(b) (f-g)(x)= (Simplify your answer.)
(c) (f⋅g)(x)= (Simplify your answer.)
(d) ((f/g))(x)= (Simplify your answer.)
(e) The domain of (f/g) is . (Type your answer in interval notation.)

Transcript text: Let $f(x)=2 x-6$ and $g(x)=8-x$. Find the following. (a) $(\mathrm{f}+\mathrm{g})(\mathrm{x})$ (b) $(f-g)(x)$ (c) $(f \cdot g)(x)$ (d) $\frac{f}{g}$ (e) The domain of $\frac{f}{g}$ (a) $(f+g)(x)=$ $\square$ (Simplify your answer.) (b) $(f-g)(x)=\square$ $\square$ (Simplify your answer.) (c) $(f \cdot g)(x)=$ $\square$ (Simplify your answer.) (d) $\left(\frac{f}{g}\right)(x)=$ $\square$ (Simplify your answer.) (e) The domain of $\frac{f}{g}$ is $\square$ . (Type your answer in interval notation.)

Solution

Solution Steps

To solve the given problems, we need to perform basic operations on the functions $ f(x) = 2x - 6 $ and $ g(x) = 8 - x $.

(a) For $(f+g)(x)$, we add the two functions: $ f(x) + g(x) $.

(b) For $(f-g)(x)$, we subtract $ g(x) $ from $ f(x) $: $ f(x) - g(x) $.

Step 1: Calculate $ (f+g)(x) $

To find $ (f+g)(x) $, we add the two functions: \[ f(x) + g(x) = (2x - 6) + (8 - x) = 2x - 6 + 8 - x = x + 2 \] Thus, \[ (f+g)(x) = x + 2 \]

Step 2: Calculate $ (f-g)(x) $

Next, we calculate $ (f-g)(x) $ by subtracting $ g(x) $ from $ f(x) $: \[ f(x) - g(x) = (2x - 6) - (8 - x) = 2x - 6 - 8 + x = 3x - 14 \] Therefore, \[ (f-g)(x) = 3x - 14 \]

Step 3: Calculate $ (f \cdot g)(x) $

Now, we find $ (f \cdot g)(x) $ by multiplying the two functions: \[ f(x) \cdot g(x) = (2x - 6)(8 - x) = 2x \cdot 8 - 2x \cdot x - 6 \cdot 8 + 6 \cdot x = 16x - 2x^2 - 48 + 6x = -2x^2 + 22x - 48 \] This can be factored as: \[ (f \cdot g)(x) = -2(x - 8)(x - 3) \]