Question

difference quotient
$\frac{f(x + h)-f(x)}{h}$
$f(x)=8x^{2}-9x$
d) 16x + 8h - 9
e) 16x - 1
f) none of the above.

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)=8x^{2}-9x$.
$f(x + h)=8(x + h)^{2}-9(x + h)=8(x^{2}+2xh+h^{2})-9x - 9h=8x^{2}+16xh+8h^{2}-9x - 9h$

Step2: Calculate $f(x + h)-f(x)$

$f(x + h)-f(x)=(8x^{2}+16xh+8h^{2}-9x - 9h)-(8x^{2}-9x)$
$=8x^{2}+16xh+8h^{2}-9x - 9h - 8x^{2}+9x=16xh+8h^{2}-9h$

Step3: Calculate the difference - quotient

$\frac{f(x + h)-f(x)}{h}=\frac{16xh+8h^{2}-9h}{h}$
Factor out $h$ from the numerator: $\frac{h(16x + 8h-9)}{h}=16x + 8h-9$

Answer:

d. $16x + 8h-9$