Use the function to evaluate the indicated expressions and simplify

Question

f(x) =3x^2+5

what is

f(x+2)=

f(x)+f(2)=

Answer 1

f(x) = 3x^2 +5

f(x+2) = 3(x+2)^2 + 5 = 3(x^2 + 4x +4) + 5 = 3x^2 + 12x + 12 + 5

= 3x^2 + 12x + 17

While if x=2, then

f(x) + f(2) = f(2) + f(2) = 2f(2)

= 2(3(2)^2 12(2)+ 5)

= 2(12+ 5) = 2(17) = 34

BUT if the x in f(x) isn't the same as the x=2 in f(2) which seems likely

f(x+2) = f(x) + f(2) when

3x^2 + 12x + 17 = 3x^2 + 22

12x = 22-17 = 5

x = 5/12

f(x+2) = f(x) + f(2) when x = 5/12

3(5/12)^2 + 12(5/12) +17

= 75/144 + 5 + 17

= 3243/144 = about 22.5

3(5/12)^2 +22

= 75/144 + 3168/144

= 3243/144 = about 22.5

then f(x) + f(2)

= 3x^2 + 5 + f(2)

= 3x^2 +5 + 3(2)^2 + 5

= 3x^2 + 5 + 17

= 3x^2 + 22

f(x) + f(2) then differs from f(x+2) by 12x -5, close but still different if x=2

buy f(x+2) = f(x) +f(2) if x= 5/12