Question

using the function, $f(x)=5-x$, find the following:
(a) $f(x+h)$
(b) $f(x+h)-f(x)$
(c) $\frac{f(x+h)-f(x)}{h}$

$f(x+h)=\square$

Explanation:

Step1: Substitute $x+h$ into $f(x)$

$f(x+h) = 5 - (x+h)$

Step2: Simplify the expression

$f(x+h) = 5 - x - h$

Step3: Compute $f(x+h)-f(x)$

$$\begin{align} f(x+h)-f(x) &= (5 - x - h) - (5 - x) \\ &= 5 - x - h - 5 + x \\ &= -h \end{align}$$

Step4: Compute the difference quotient

$$\begin{align} \frac{f(x+h)-f(x)}{h} &= \frac{-h}{h} \\ &= -1 \end{align}$$

Answer:

(a) $5 - x - h$
(b) $-h$
(c) $-1$