Questions: Given that f(x)=(x-4)^2 and g(x)=7-2x, find (a) (f+g)(x)= (b) (f-g)(x)= (c) (fg)(x)= (d) (f/g)(x)=

Transcript text: Given that $f(x)=(x-4)^{2}$ and $g(x)=7-2 x$, find (a) $(f+g)(x)=$ $\square$ (b) $(f-g)(x)=$ $\square$ (c) $(f g)(x)=$ $\square$ (d) $\left(\frac{f}{g}\right)(x)=$ $\square$

Solution

Solution Steps

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

(a) For $(f+g)(x)$, we add the expressions for $ f(x) $ and $ g(x) $.

(b) For $(f-g)(x)$, we subtract the expression for $ g(x) $ from $ f(x) $.

Step 1: Define the Functions

We are given two functions: \[ f(x) = (x - 4)^2 \] \[ g(x) = 7 - 2x \]

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

To find $(f+g)(x)$, we add the expressions for $ f(x) $ and $ g(x) $: \[ (f+g)(x) = (x - 4)^2 + (7 - 2x) \] Expanding this expression, we get: \[ (f+g)(x) = x^2 - 10x + 23 \]

Step 3: Calculate $(f-g)(x)$

To find $(f-g)(x)$, we subtract the expression for $ g(x) $ from $ f(x) $: \[ (f-g)(x) = (x - 4)^2 - (7 - 2x) \] Expanding this expression, we get: \[ (f-g)(x) = x^2 - 6x + 9 \]

Step 4: Calculate $(fg)(x)$

To find $(fg)(x)$, we multiply the expressions for $ f(x) $ and $ g(x) $: \[ (fg)(x) = (x - 4)^2 \cdot (7 - 2x) \] Expanding this expression, we get: \[ (fg)(x) = -2x^3 + 23x^2 - 88x + 112 \]