Questions: Find the simplified expression for f(5+h) for the following function, f(x) = 3x^2 - 2x + 4

Transcript text: Find the simplified expression for $f(5+h)$ for the following function, \[ f(x)=3 x^{2}-2 x+4 \]

Solution

Solution Steps

To find the simplified expression for $ f(5+h) $ given the function $ f(x) = 3x^2 - 2x + 4 $, we need to substitute $ 5+h $ for $ x $ in the function and then simplify the resulting expression.

Solution Approach

Substitute $ 5+h $ into the function $ f(x) $.
Expand the expression.
Combine like terms to simplify.

Step 1: Substitute $ 5+h $ into the function $ f(x) $

Given the function $ f(x) = 3x^2 - 2x + 4 $, we substitute $ x $ with $ 5+h $: \[ f(5+h) = 3(5+h)^2 - 2(5+h) + 4 \]

Step 2: Expand the expression

Next, we expand the expression: \[ f(5+h) = 3(25 + 10h + h^2) - 2(5 + h) + 4 \] \[ = 3 \cdot 25 + 3 \cdot 10h + 3 \cdot h^2 - 2 \cdot 5 - 2 \cdot h + 4 \] \[ = 75 + 30h + 3h^2 - 10 - 2h + 4 \]

Step 3: Combine like terms

Now, we combine like terms to simplify the expression: \[ f(5+h) = 3h^2 + 30h - 2h + 75 - 10 + 4 \] \[ = 3h^2 + 28h + 69 \]