Questions: f(x) = 7x^2 + 5 F(x+2) = 7(x+2)^2 + 5

Transcript text: $\begin{aligned} f(x) & =7 x^{2}+5 \\ F(x+2) & =7(x+2)^{2}+5\end{aligned}$

Solution

Solution Steps

To solve this problem, we need to evaluate the function $ f(x) = 7x^2 + 5 $ at $ x+2 $. This involves substituting $ x+2 $ into the function in place of $ x $ and simplifying the expression.

Step 1: Define the Function

The function given is $ f(x) = 7x^2 + 5 $.

Step 2: Substitute $ x+2 $ into the Function

To find $ F(x+2) $, substitute $ x+2 $ into the function: \[ F(x+2) = 7(x+2)^2 + 5 \]

Step 3: Expand the Expression

Expand the expression $ (x+2)^2 $: \[ (x+2)^2 = x^2 + 4x + 4 \]

Step 4: Simplify the Function

Substitute the expanded expression back into the function: \[ F(x+2) = 7(x^2 + 4x + 4) + 5 \] Distribute the 7: \[ F(x+2) = 7x^2 + 28x + 28 + 5 \] Combine like terms: \[ F(x+2) = 7x^2 + 28x + 33 \]

Step 5: Evaluate at $ x = 0 $

Substitute $ x = 0 $ into the simplified function: \[ F(0+2) = 7(0)^2 + 28(0) + 33 = 33 \]