Question

For the function f(x) = 3x² + 4x – 8, evaluate and fully simplify each of the following. f(x + h) (f(x + h) – f(x))/h

a) 3x² + 4x + h
b) 9x + 4h – 8
c) 3x² + 4x + 4h
d) 6x + 4

Answer 1

To evaluate f(x + h), substitute (x + h) into the function f(x) = 3x² + 4x – 8. To evaluate (f(x + h) – f(x))/h, substitute f(x + h) and f(x) into the expression and simplify.

Step-by-step explanation:

To evaluate f(x + h), we substitute (x + h) into the function f(x) = 3x² + 4x – 8:
f(x + h) = 3(x + h)² + 4(x + h) – 8
Expanding and simplifying:
f(x + h) = 3(x² + 2hx + h²) + 4x + 4h – 8
Distributing brackets and combining like terms:
f(x + h) = 3x² + 6hx + 3h² + 4x + 4h – 8

To evaluate (f(x + h) – f(x))/h, we substitute f(x + h) and f(x) into the expression and simplify:
(f(x + h) – f(x))/h = (3x² + 6hx + 3h² + 4x + 4h – 8 – (3x² + 4x – 8))/h
Simplifying:
(f(x + h) – f(x))/h = (6hx + 3h² + 4h)/h
Factoring out h:
(f(x + h) – f(x))/h = h(6x + 3h + 4)/h
Simplifying further:
(f(x + h) – f(x))/h = 6x + 3h + 4