Question

find and simplify the difference quotient \\(\frac{f(x + h) - f(x)}{h}\\), \\(h \
eq 0\\) for the given function.\\(f(x) = -3x\\)\\(\frac{f(x + h) - f(x)}{h} = \square\\) (simplify your answer.)

Explanation:

Step1: Find $f(x+h)$

Substitute $x+h$ into $f(x)$:
$f(x+h) = -3(x+h) = -3x -3h$

Step2: Substitute into difference quotient

Plug $f(x+h)$ and $f(x)$ into $\frac{f(x+h)-f(x)}{h}$:
$\frac{(-3x -3h) - (-3x)}{h}$

Step3: Simplify the numerator

Combine like terms in the numerator:
$\frac{-3x -3h +3x}{h} = \frac{-3h}{h}$

Step4: Cancel common factor $h$

Since $h
eq0$, divide numerator and denominator by $h$:
$\frac{-3h}{h} = -3$

Answer:

$-3$