Let's construct and simplify the difference quotient for the function [tex]\( f(x) = x^2 - 9 \)[/tex].
1. Function Evaluation:
- [tex]\( f(x) = x^2 - 9 \)[/tex]
- [tex]\( f(x + h) = (x + h)^2 - 9 \)[/tex]
2. Calculate [tex]\( f(x + h) - f(x) \)[/tex]:
- [tex]\( f(x + h) - f(x) = [(x + h)^2 - 9] - [x^2 - 9] \)[/tex]
3. Simplify the Expression:
- Expand [tex]\( (x + h)^2 \)[/tex]:
[tex]\[
(x + h)^2 = x^2 + 2xh + h^2
\][/tex]
- So,
[tex]\[
f(x + h) = x^2 + 2xh + h^2 - 9
\][/tex]
- Subtract [tex]\( f(x) = x^2 - 9 \)[/tex]:
[tex]\[
f(x + h) - f(x) = (x^2 + 2xh + h^2 - 9) - (x^2 - 9)
\][/tex]
- This simplifies to:
[tex]\[
f(x + h) - f(x) = x^2 + 2xh + h^2 - 9 - x^2 + 9 = 2xh + h^2
\][/tex]
4. Construct the Difference Quotient:
- The difference quotient is:
[tex]\[
\frac{f(x + h) - f(x)}{h} = \frac{2xh + h^2}{h}
\][/tex]
5. Simplify the Difference Quotient:
- Factor out [tex]\( h \)[/tex] in the numerator:
[tex]\[
\frac{2xh + h^2}{h} = \frac{h(2x + h)}{h}
\][/tex]
- Cancel the common factor [tex]\( h \)[/tex]:
[tex]\[
\frac{f(x + h) - f(x)}{h} = 2x + h
\][/tex]
Thus, the simplified difference quotient for the function [tex]\( f(x) = x^2 - 9 \)[/tex] is:
[tex]\[
2x + h
\][/tex]
Therefore, the difference quotient is [tex]\( h + 2x \)[/tex].
