Question

find each of the following for f(x) = 3x² - 8x + 9. (a) f(x + h) (b) f(x + h) - f(x) (c) \\(\frac{f(x + h) - f(x)}{h}\\) (a) f(x + h) = \\(\square\\)

Explanation:

Step1: Substitute x+h into f(x)

$f(x+h)=3(x+h)^2 -8(x+h)+9=3(x^2+2xh+h^2)-8x-8h+9=3x^2+6xh+3h^2-8x-8h+9$

Step2: Compute f(x+h)-f(x)

$f(x+h)-f(x)=(3x^2+6xh+3h^2-8x-8h+9)-(3x^2-8x+9)=6xh+3h^2-8h$

Step3: Divide by h

$\frac{f(x+h)-f(x)}{h}=\frac{6xh+3h^2-8h}{h}=6x+3h-8$

Answer:

(a) $3x^2 + 6xh + 3h^2 - 8x - 8h + 9$
(b) $6xh + 3h^2 - 8h$
(c) $6x + 3h - 8$