Question

find the difference quotient of f, that is, find \\( \frac { f ( x + h ) - f ( x ) } { h } \\), \\( h \
eq 0 \\), for the following function. \\( f ( x ) = 8 x + 5 \\) \\( \frac { f ( x + h ) - f ( x ) } { h } = \\) (simplify your answer.)

Explanation:

Step1: Find \( f(x + h) \)

Given \( f(x)=8x + 5 \), substitute \( x\) with \(x + h\) in the function.
\( f(x + h)=8(x + h)+5=8x+8h + 5 \)

Step2: Substitute \( f(x + h) \) and \( f(x) \) into the difference - quotient formula

The difference - quotient formula is \(\frac{f(x + h)-f(x)}{h}\).
Substitute \( f(x + h)=8x + 8h+5\) and \( f(x)=8x + 5\) into it:

$$
LATEXBLOCK0
$$

Step3: Simplify the numerator and then the fraction

Simplify the numerator: \(8x+8h + 5-8x - 5 = 8h\)
So, \(\frac{8h}{h}\), and since \(h
eq0\), we can cancel out the \(h\) terms.

Answer:

\(8\)