To evaluate [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex] given that [tex]\(f(x) = x^2 - 5\)[/tex], follow these steps:
1. Determine [tex]\(f(x)\)[/tex]:
Given [tex]\(f(x) = x^2 - 5\)[/tex].
2. Determine [tex]\(f(x+h)\)[/tex]:
Substitute [tex]\(x + h\)[/tex] into the function:
[tex]\[
f(x+h) = (x + h)^2 - 5
\][/tex]
3. Expand [tex]\(f(x+h)\)[/tex]:
[tex]\[
f(x+h) = (x + h)^2 - 5
= x^2 + 2xh + h^2 - 5
\][/tex]
4. Substitute [tex]\(f(x)\)[/tex] and [tex]\(f(x+h)\)[/tex] into the difference quotient:
The difference quotient is:
[tex]\[
\frac{f(x+h) - f(x)}{h}
\][/tex]
Plug in [tex]\(f(x)\)[/tex] and [tex]\(f(x+h)\)[/tex]:
[tex]\[
\frac{(x^2 + 2xh + h^2 - 5) - (x^2 - 5)}{h}
\][/tex]
5. Simplify the numerator:
[tex]\[
(x^2 + 2xh + h^2 - 5) - (x^2 - 5)
= x^2 + 2xh + h^2 - 5 - x^2 + 5
= 2xh + h^2
\][/tex]
6. Combine like terms:
[tex]\[
\frac{2xh + h^2}{h}
\][/tex]
7. Factor out [tex]\(h\)[/tex] from the numerator:
[tex]\[
\frac{h(2x + h)}{h}
\][/tex]
8. Cancel the [tex]\(h\)[/tex] in the numerator and the denominator:
[tex]\[
2x + h
\][/tex]
Therefore, the evaluated difference quotient [tex]\(\frac{f(x+h) - f(x)}{h}\)[/tex] is:
[tex]\[
\frac{f(x+h) - f(x)}{h} = \frac{(x^2 + 2xh + h^2 - 5) - (x^2 - 5)}{h} = 2x + h
\][/tex]
