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

Question

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

Kennedy Adams · Accepted Answer

#f(x+2)-f(x)/2=x^2+8x+17/2#

#f(x) = 2x^2 + 1#

#f(x+2) - f(x)/2#

First, find #f(x+2)# by plugging #x+2# into the original equation

#f(x+2) = 2(x+2)^2 + 1# #f(x+2) = 2(x^2+4x+4) + 1# #f(x+2) = 2x^2+8x+9#

Next, find #f(x)/2# by substituting #2x^2+1# for #f(x)#

#f(x)/2 = (2x^2+1)/2# #f(x)/2 = x^2+1/2#

For the final response, combine your responses via substitution.

#f(x+2) - f(x)/2# #= (2x^2+8x+9) - (x^2+1/2)# #=x^2+8x+17/2#

Kennedy Adams · Answer

#x^2+8x+17/2# #f(x+2)-f(x)/2# #" "# #=2(x+2)^2+1-(2x^2+1)/2# #" "# #=2(x^2+4x+4)+1-(2x^2)/2-1/2# #" "# #=2x^2+8x+8+1-x^2-1/2# #" "# #=2x^2-x^2+8x+8+1-1/2# #" "# #=x^2+8x+17/2#

Nathan Axel · Answer

undefined To find and simplify $ \frac{{f(x+2) - f(x)}}{2} $, where $ f(x) = 2x^2 + 1 $:

1. Substitute $ x+2 $ and $ x $ into $ f(x) $ to get $ f(x+2) $ and $ f(x) $.
2. Calculate $ f(x+2) - f(x) $.
3. Divide the result by 2.
4. Simplify the expression obtained.

Let's proceed:

1. $ f(x+2) = 2(x+2)^2 + 1 = 2(x^2 + 4x + 4) + 1 = 2x^2 + 8x + 8 + 1 = 2x^2 + 8x + 9 $.
   
   $ f(x) = 2x^2 + 1 $.

2. $ f(x+2) - f(x) = (2x^2 + 8x + 9) - (2x^2 + 1) = 8x + 9 - 1 = 8x + 8 $.

3. $ \frac{{f(x+2) - f(x)}}{2} = \frac{{8x + 8}}{2} = 4x + 4 $.

Therefore, $ \frac{{f(x+2) - f(x)}}{2} = 4x + 4 $.