Answer:
a) f(x + h) = (x + h)^2 + 3(x + h) + 2
b) f(x + h) - f(x) = h^2 + 2xh + 3h
c) f(x + h) - f(x)) ÷ h = 2x + h + 3
Explanation:
Writing f(x) = x^2 + 3x + 2 means that within this equation, 'x' is defined as a variable. So for the question f(x + h), we can replace all the x's with x + h, just as you would replace all the x's with a 5 if it asked for f(5). Hence:
f(x + h) = (x + h)^2 + 3(x + h) + 2 ← this is the most simplified version
f(x + h) = x^2 + 2xh + h^2 + 3x + 3h + 2 ← expanded version
∴ f(x + h) = (x + h)^2 + 3(x + h) + 2
f(x + h) - f(x) = (x^2 + 2xh + h^2 + 3x + 3h + 2 ) - (x^2 + 3x +2) ← using already expanded version from above
f(x + h) - f(x) =x^2 + 2xh + h^2 + 3x + 3h + 2 - x^2 - 3x - 2 ← expanding brackets
f(x + h) - f(x) = 2xh + h^2 + 3h ← simplifying common terms
∴ f(x + h) - f(x) = h^2 + 2xh + 3h
(f(x + h) - f(x)) ÷ h = (2xh + h^2 + 3h) ÷ h ← taking answer from above
(f(x + h) - f(x)) ÷ h = (2x + h + 3) ← dividing each term by h
∴ (f(x + h) - f(x)) ÷ h = 2x + h + 3
This type of problem will become the basis of finding derivatives through natural principles in calculus :) Hope that helps!