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 + 3 \\)
\\( \frac { f ( x + h ) - f ( x ) } { h } = \square \\) (simplify your answer.)

Explanation:

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

Given \( f(x)=8x + 3 \), substitute \( x\) with \(x + h\) in \(f(x)\).
\(f(x + h)=8(x + h)+3=8x+8h + 3\)

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

\(\frac{f(x + h)-f(x)}{h}=\frac{(8x + 8h+3)-(8x + 3)}{h}\)

Step3: Simplify the numerator

\((8x + 8h+3)-(8x + 3)=8x+8h + 3-8x - 3=8h\)

Step4: Simplify the fraction

\(\frac{8h}{h}=8\) (since \(h
eq0\), we can cancel out the \(h\) terms)

Answer:

\(8\)