Question

given the function $g(x) = 5 - x^2$, evaluate $\frac{f(x + h) - f(x)}{h}$, where $h \
eq 0$. select one: \LXI0 a. $-2x - h$ \LXI1 b. $-2x + h$ \LXI2 c. $-2x - h^2$ \LXI3 d. $2x + h$

Explanation:

Step1: Define $g(x+h)$

Substitute $x+h$ into $g(x)$:
$g(x+h)=5-(x+h)^2=5-(x^2+2xh+h^2)=5-x^2-2xh-h^2$

Step2: Compute $g(x+h)-g(x)$

Subtract $g(x)$ from $g(x+h)$:
$g(x+h)-g(x)=(5-x^2-2xh-h^2)-(5-x^2)=-2xh-h^2$

Step3: Divide by $h$

Factor and simplify the difference quotient:
$\frac{g(x+h)-g(x)}{h}=\frac{h(-2x-h)}{h}=-2x-h$ (since $h
eq0$)

Answer:

A. $-2x - h$