Question

find the difference quotient of f; that is, find \\( \frac { f ( x + h ) - f ( x ) } { h } \\), \\( h \
eq 0 \\), for the following function. be sure to simplify. \\( f ( x ) = x ^ { 2 } - 2 x + 6 \\) \\( \frac { f ( x + h ) - f ( x ) } { h } = \square \\) (simplify your answer.)

Explanation:

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

Substitute \( x + h \) into \( f(x) \):
\( f(x + h)=(x + h)^2-2(x + h)+6=x^{2}+2xh+h^{2}-2x - 2h+6 \)

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

\( f(x + h)-f(x)=(x^{2}+2xh+h^{2}-2x - 2h+6)-(x^{2}-2x + 6) \)
\( =x^{2}+2xh+h^{2}-2x - 2h+6 - x^{2}+2x - 6 \)
\( =2xh+h^{2}-2h \)

Step3: Compute \( \frac{f(x + h)-f(x)}{h} \)

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

Answer:

\( 2x+h - 2 \)