Part (a): For the expression f(x - 1) + f(x + 1) = 18x^2 + 20. Part (b): For the expression f(x + 2h) = 9x^2 + 35hx + 36h^2 - 2h + 1.
Let's find the following for the function f(x) = 9x^2 – x + 1:
(a) f(x - 1) + f(x + 1)
(b) f(x + 2h)
Steps to solve:
Part (a):
Distribute the terms:
f(x - 1) + f(x + 1) = 9(x - 1)^2 - (x - 1) + 9(x + 1)^2 + (x + 1)
Expand the squares:
f(x - 1) + f(x + 1) = 9(x^2 - 2x + 1) - x + 1 + 9(x^2 + 2x + 1) + x + 1
Combine like terms:
f(x - 1) + f(x + 1) = 18x^2 + 18 + 2
Simplify:
f(x - 1) + f(x + 1) = 18x^2 + 20
Part (b):
Substitute x + 2h into the function:
f(x + 2h) = 9(x + 2h)^2 - (x + 2h) + 1
Expand the square:
f(x + 2h) = 9(x^2 + 4hx + 4h^2) - x - 2h + 1
Distribute the terms:
f(x + 2h) = 9x^2 + 36hx + 36h^2 - x - 2h + 1
Combine like terms:
f(x + 2h) = 9x^2 + 35hx + 36h^2 - 2h + 1
Complete question:
Let f(x) = 9x^2 – x + 1. Find the following. = (a) f(x - 1) + f(x + 1) (b) f(x + 2h)