Question

find the difference quotient of f, that is, find \\(\frac{f(x + h) - f(x)}{h}\\), \\(h \
eq 0\\), for the following function. be sure to simplify \\(f(x) = x^2 - 2x + 4\\) \\(\frac{f(x + h) - f(x)}{h} = \square\\) (simplify your answer)

Explanation:

Step1: Compute $f(x+h)$

Expand $(x+h)^2 - 2(x+h) + 4$:
$$f(x+h)=(x+h)^2 - 2(x+h) + 4 = x^2 + 2xh + h^2 - 2x - 2h + 4$$

Step2: Substitute into difference quotient

Plug $f(x+h)$ and $f(x)$ into $\frac{f(x+h)-f(x)}{h}$:
$$\frac{(x^2 + 2xh + h^2 - 2x - 2h + 4) - (x^2 - 2x + 4)}{h}$$

Step3: Simplify the numerator

Cancel like terms in the numerator:
$$\frac{x^2 + 2xh + h^2 - 2x - 2h + 4 - x^2 + 2x - 4}{h} = \frac{2xh + h^2 - 2h}{h}$$

Step4: Factor and cancel $h$

Factor $h$ from the numerator and divide by $h$ ($h
eq0$):
$$\frac{h(2x + h - 2)}{h} = 2x + h - 2$$

Answer:

$2x + h - 2$