I need help solving this problem f(x)=2x^2+1 find and simplify f(x+2)-f(x)/2?

Feb 2, 2017

$f \left(x + 2\right) - f \frac{x}{2} = {x}^{2} + 8 x + \frac{17}{2}$

Explanation:

$f \left(x\right) = 2 {x}^{2} + 1$

$f \left(x + 2\right) - f \frac{x}{2}$

First, find $f \left(x + 2\right)$ by plugging $x + 2$ into the original equation

$f \left(x + 2\right) = 2 {\left(x + 2\right)}^{2} + 1$
$f \left(x + 2\right) = 2 \left({x}^{2} + 4 x + 4\right) + 1$
$f \left(x + 2\right) = 2 {x}^{2} + 8 x + 9$

Next, find $f \frac{x}{2}$ by substituting $2 {x}^{2} + 1$ for $f \left(x\right)$

$f \frac{x}{2} = \frac{2 {x}^{2} + 1}{2}$
$f \frac{x}{2} = {x}^{2} + \frac{1}{2}$

$f \left(x + 2\right) - f \frac{x}{2}$
$= \left(2 {x}^{2} + 8 x + 9\right) - \left({x}^{2} + \frac{1}{2}\right)$
$= {x}^{2} + 8 x + \frac{17}{2}$

Feb 2, 2017

${x}^{2} + 8 x + \frac{17}{2}$

Explanation:

$f \left(x + 2\right) - f \frac{x}{2}$
$\text{ }$
$= 2 {\left(x + 2\right)}^{2} + 1 - \frac{2 {x}^{2} + 1}{2}$
$\text{ }$
$= 2 \left({x}^{2} + 4 x + 4\right) + 1 - \frac{2 {x}^{2}}{2} - \frac{1}{2}$
$\text{ }$
$= 2 {x}^{2} + 8 x + 8 + 1 - {x}^{2} - \frac{1}{2}$
$\text{ }$
$= 2 {x}^{2} - {x}^{2} + 8 x + 8 + 1 - \frac{1}{2}$
$\text{ }$
$= {x}^{2} + 8 x + \frac{17}{2}$