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

Explanation:

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

Substitute \( x + h \) into \( f(x) \):
\( f(x + h)=(x + h)^{2}-9(x + h)+2 \)
Expand \( (x + h)^{2} \) and distribute -9:
\( f(x + h)=x^{2}+2xh+h^{2}-9x - 9h+2 \)

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

Subtract \( f(x)=x^{2}-9x + 2 \) from \( f(x + h) \):
\( f(x + h)-f(x)=(x^{2}+2xh+h^{2}-9x - 9h+2)-(x^{2}-9x + 2) \)
Simplify by canceling \( x^{2} \), -9x, and 2:
\( f(x + h)-f(x)=2xh+h^{2}-9h \)

Step3: Divide by \( h \) ( \( h

eq0 \))
\( \frac{f(x + h)-f(x)}{h}=\frac{2xh+h^{2}-9h}{h} \)
Factor out \( h \) in the numerator:
\( \frac{f(x + h)-f(x)}{h}=\frac{h(2x + h-9)}{h} \)
Cancel \( h \):
\( \frac{f(x + h)-f(x)}{h}=2x + h-9 \)

Answer:

\( 2x + h - 9 \)