Questions: Let f(x)=8x-4x^2. Find and simplify each of the following: f(x)+h-f(x)= f(x+h)-f(x)=

Transcript text: Let $f(x)=8 x-4 x^{2}$. Find and simplify each of the following: \[ \begin{array}{l} f(x)+h-f(x)= \\ f(x+h)-f(x)= \end{array} \]

Solution

Solution Steps

To solve the given problem, we need to find and simplify the expressions $ f(x) + h - f(x) $ and $ f(x+h) - f(x) $ for the function $ f(x) = 8x - 4x^2 $.

For $ f(x) + h - f(x) $:
- This expression simplifies directly to $ h $ because $ f(x) $ cancels out.
For $ f(x+h) - f(x) $:
- First, substitute $ x + h $ into the function $ f $ to get $ f(x+h) $.
- Then, subtract $ f(x) $ from $ f(x+h) $.
- Simplify the resulting expression.

Step 1: Calculate $ f(x) + h - f(x) $

Given the expression $ f(x) + h - f(x) $, we can simplify it as follows: \[ f(x) + h - f(x) = h \] Substituting $ h = 1 $: \[ f(x) + 1 - f(x) = 1 \] Thus, the result for this expression is: \[ \boxed{1} \]

Step 2: Calculate $ f(x+h) - f(x) $

Next, we need to evaluate the expression $ f(x+h) - f(x) $. First, we substitute $ x + h $ into the function $ f $: \[ f(x+h) = f(2 + 1) = f(3) = 8(3) - 4(3^2) = 24 - 36 = -12 \] Now, we calculate $ f(x) $: \[ f(x) = f(2) = 8(2) - 4(2^2) = 16 - 16 = 0 \] Now, we can find $ f(x+h) - f(x) $: \[ f(x+h) - f(x) = -12 - 0 = -12 \] Thus, the result for this expression is: \[ \boxed{-12} \]