Answer
$\dfrac{f(x+h)-f(x)}{h}=-5$
$\dfrac{f(x)-f(a)}{x-a}=-5$
Work Step by Step
$f(x)=4-5x$
$\textbf{Evaluate}$ $\dfrac{f(x+h)-f(x)}{h}$
First, substitute $x$ by $x+h$ in the given function and simplify to find $f(x+h)$:
$f(x+h)=4-5(x+h)=4-5x-5h$
Substitute $f(x+h)$ and $f(x)$ into the difference quotient formula and simplify:
$\dfrac{f(x+h)-f(x)}{h}=\dfrac{4-5x-5h-(4-5x)}{h}=...$
$...=\dfrac{4-5x-5h-4+5x}{h}=\dfrac{-5h}{h}=-5$
$\textbf{Evaluate}$ $\dfrac{f(x)-f(a)}{x-a}$
Substitute $x$ by $a$ in $f(x)$ to find $f(a)$:
$f(a)=4-5a$
Substitute $f(a)$ and $f(x)$ into the difference quotient formula and simplify:
$\dfrac{f(x)-f(a)}{x-a}=\dfrac{4-5x-(4-5a)}{x-a}=\dfrac{4-5x-4+5a}{x-a}=...$
$...=\dfrac{-5x+5a}{x-a}=...$
Take out common factor $-5$ from the numerator and simplify again:
$...=\dfrac{-5(x-a)}{x-a}=-5$
