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 ) = x ^ { 2 } - 9 x + 3 \\)
\\( \frac { f ( x + h ) - f ( x ) } { h } = \\) (simplify your answer.)

Explanation:

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

Substitute \( x + h \) into \( f(x)=x^{2}-9x + 3 \).
\( f(x + h)=(x + h)^{2}-9(x + h)+3=x^{2}+2xh+h^{2}-9x-9h + 3 \)

Step2: Calculate \( f(x + h)-f(x) \)

\( f(x + h)-f(x)=(x^{2}+2xh+h^{2}-9x-9h + 3)-(x^{2}-9x + 3) \)

$$
LATEXBLOCK0
$$

Step3: Find the difference quotient \(\frac{f(x + h)-f(x)}{h}\)

\(\frac{f(x + h)-f(x)}{h}=\frac{2xh+h^{2}-9h}{h}\), since \(h
eq0\), we can factor out \(h\) in the numerator:
\(\frac{h(2x + h-9)}{h}=2x+h - 9\)

Answer:

\(2x+h - 9\)