Math Problem Statement

difference quotient of f(x) = (x-1)/(x+2)

Solution

To find the difference quotient of the function f(x)=x1x+2f(x) = \frac{x - 1}{x + 2}, we'll use the formula for the difference quotient:

f(x+h)f(x)h\frac{f(x+h) - f(x)}{h}

Step-by-step process:

  1. Find f(x+h)f(x+h): Substitute x+hx+h into f(x)f(x): f(x+h)=(x+h)1(x+h)+2=x+h1x+h+2f(x+h) = \frac{(x+h) - 1}{(x+h) + 2} = \frac{x+h-1}{x+h+2}

  2. Compute f(x+h)f(x)f(x+h) - f(x): Now, subtract f(x)f(x) from f(x+h)f(x+h): f(x+h)f(x)=x+h1x+h+2x1x+2f(x+h) - f(x) = \frac{x+h-1}{x+h+2} - \frac{x-1}{x+2}

    To subtract these fractions, first find a common denominator. The common denominator is (x+h+2)(x+2)(x+h+2)(x+2). So, we have:

    f(x+h)f(x)=(x+h1)(x+2)(x1)(x+h+2)(x+h+2)(x+2)f(x+h) - f(x) = \frac{(x+h-1)(x+2) - (x-1)(x+h+2)}{(x+h+2)(x+2)}

    Expanding both numerators: (x+h1)(x+2)=x2+2x+hx+2hx2(x+h-1)(x+2) = x^2 + 2x + hx + 2h - x - 2 Simplifying: =x2+hx+(2h1)x2= x^2 + hx + (2h - 1)x - 2

    Similarly, expand (x1)(x+h+2)(x-1)(x+h+2): (x1)(x+h+2)=x2+hx+2xxh2(x-1)(x+h+2) = x^2 + hx + 2x - x - h - 2 Simplifying: =x2+hx+xh2= x^2 + hx + x - h - 2

    Now, subtract the second numerator from the first: (x2+hx+(2h1)x2)(x2+hx+xh2)=(2h1)xx+h=(2h2)x+h(x^2 + hx + (2h-1)x - 2) - (x^2 + hx + x - h - 2) = (2h-1)x - x + h = (2h-2)x + h

  3. Simplify the difference quotient expression: Now, we can express f(x+h)f(x)f(x+h) - f(x) as: f(x+h)f(x)=(2h2)x+h(x+h+2)(x+2)f(x+h) - f(x) = \frac{(2h-2)x + h}{(x+h+2)(x+2)}

  4. Divide by hh: Finally, divide by hh: f(x+h)f(x)h=(2h2)x+hh(x+h+2)(x+2)\frac{f(x+h) - f(x)}{h} = \frac{(2h-2)x + h}{h(x+h+2)(x+2)} Simplifying further: =(2h2)x+hh(x+h+2)(x+2)=(2x2)+1(x+h+2)(x+2)=2x1(x+h+2)(x+2)= \frac{(2h-2)x + h}{h(x+h+2)(x+2)} = \frac{(2x-2) + 1}{(x+h+2)(x+2)} = \frac{2x-1}{(x+h+2)(x+2)}

Thus, the difference quotient of f(x)=x1x+2f(x) = \frac{x-1}{x+2} is:

2x1(x+h+2)(x+2)\frac{2x-1}{(x+h+2)(x+2)}

Would you like more details on any step or a further breakdown?


Here are 5 related questions for further practice:

  1. What is the derivative of f(x)=x1x+2f(x) = \frac{x-1}{x+2}?
  2. How does the difference quotient relate to the derivative?
  3. What happens to the difference quotient as h0h \to 0?
  4. How do you simplify difference quotients with more complex functions?
  5. What is the difference quotient of f(x)=1xf(x) = \frac{1}{x}?

Tip: The difference quotient is essentially the slope of the secant line between two points on the curve, and it approaches the derivative as the distance between the points shrinks (as h0h \to 0).

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Calculus
Difference Quotient
Algebra
Rational Functions

Formulas

Difference Quotient: (f(x+h) - f(x)) / h

Theorems

Definition of Derivative
Properties of Rational Functions

Suitable Grade Level

Grades 11-12 (Pre-Calculus/Calculus)