Problem 89 - Precalculus Essentials

Question

Find and simplify the difference quotient $$\frac{f(x+h)-f(x)}{h}, h \neq 0$$for the given function. $$f(x)=\frac{1}{x}$$

Step-by-Step Solution

Answer

The simplified difference quotient for \(f(x) = \frac{1}{x}\) is \(\frac{-1}{x(x+h)}\)

Step 1: Substitute the Function into the Difference Quotient

Plug the function \(f(x) = \frac{1}{x}\) into the difference quotient \(\frac{f(x+h)-f(x)}{h}\). This, yields to \(\frac{f(x+h)-f(x)}{h} = \frac{\frac{1}{x+h}-\frac{1}{x}}{h}\)

Step 2: Create a Common Denominator

As with fractions, when subtracting fractions, a common denominator is needed. Multiply the first fraction by \(\frac{x}{x}\) and the second fraction by \(\frac{x+h}{x+h}\), this yields to \(\frac{x-(x+h)}{hx(x+h)}\)

Step 3: Simplify the Numerator

Simplify the numerator of the fraction by subtracting x from (x+h), which give us \(h\) in the numerator. So we get \(\frac{h}{hx(x+h)}\)

Step 4: Simplify the Difference Quotient Expression

Recall that the difference quotient is defined for \(h \neq 0\), so we can safely cancel the \(h\) in the numerator with one \(h\) in the denominator. This gives the final answer of \(\frac{-1}{x(x + h)}\)