Horton Mullins
04/14/2023 · Junior High School
Given the function \( f(x)=\frac{1}{x} \), find the difference quotient \( \frac{f(x+h)-f(x)}{h} \). \( \frac{f(x+h)-f(x)}{h}= \) (Enter the numerator and denominator separately in
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Step-by-step Solution
To find the difference quotient for the function \( f(x) = \frac{1}{x} \), we start by calculating \( f(x+h) \) and \( f(x) \).
1. Calculate \( f(x+h) \):
\[
f(x+h) = \frac{1}{x+h}
\]
2. Calculate \( f(x) \):
\[
f(x) = \frac{1}{x}
\]
3. Now, substitute these into the difference quotient:
\[
\frac{f(x+h) - f(x)}{h} = \frac{\frac{1}{x+h} - \frac{1}{x}}{h}
\]
4. To combine the fractions in the numerator, find a common denominator:
\[
\frac{1}{x+h} - \frac{1}{x} = \frac{x - (x+h)}{x(x+h)} = \frac{x - x - h}{x(x+h)} = \frac{-h}{x(x+h)}
\]
5. Substitute this back into the difference quotient:
\[
\frac{f(x+h) - f(x)}{h} = \frac{\frac{-h}{x(x+h)}}{h}
\]
6. Simplify the expression:
\[
= \frac{-h}{h \cdot x(x+h)} = \frac{-1}{x(x+h)} \quad \text{(for } h \neq 0\text{)}
\]
Thus, the difference quotient is:
\[
\frac{f(x+h) - f(x)}{h} = \frac{-1}{x(x+h)}
\]
### Final Answer:
- **Numerator**: \(-1\)
- **Denominator**: \(x(x+h)\)
Quick Answer
Numerator: -1, Denominator: \(x(x+h)\)
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