Find the Difference Quotient f(x)=(3-2x)/x

Problem

ƒ(x)=(3−2*x)/x

Solution

1. Identify the difference quotient formula, which is defined as (ƒ*(x+h)−ƒ(x))/h for h≠0

2. Evaluate ƒ*(x+h) by substituting x+h for every x in the original function.

ƒ*(x+h)=(3−2*(x+h))/(x+h)

3. Simplify the expression for ƒ*(x+h) by distributing the −2 in the numerator.

ƒ*(x+h)=(3−2*x−2*h)/(x+h)

4. Substitute the expressions for ƒ*(x+h) and ƒ(x) into the numerator of the difference quotient.

(3−2*x−2*h)/(x+h)−(3−2*x)/x

5. Find a common denominator for the terms in the numerator, which is x*(x+h)

(x*(3−2*x−2*h)−(3−2*x)*(x+h))/(x*(x+h))

6. Expand the terms in the numerator to combine like terms.

(3*x−2*x2−2*x*h−(3*x+3*h−2*x2−2*x*h))/(x*(x+h))

7. Distribute the negative sign and simplify the numerator.

(3*x−2*x2−2*x*h−3*x−3*h+2*x2+2*x*h)/(x*(x+h))

(−3*h)/(x*(x+h))

8. Divide the simplified numerator by h to complete the difference quotient formula.

(−3*h)/(x*(x+h))/h

9. Cancel the common factor of h from the numerator and denominator.

(−3)/(x*(x+h))

Final Answer

(ƒ*(x+h)−ƒ(x))/h=(−3)/(x*(x+h))

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