Math Problem Statement

Given the function f(x) = (x^2 - 2x) / x. a) Find the domain A of the function. b) Prove that for all x ∈ A: f(x) = x - 2. c) Compute the limit lim (x → 0) f(x).

Solution

The given mathematical exercise consists of the function f(x)=x22xxf(x) = \frac{x^2 - 2x}{x}. Let's solve the three parts step by step:

a) Finding the domain AA of the function

The function is defined as:

f(x)=x22xxf(x) = \frac{x^2 - 2x}{x}

We know that the function is not defined when the denominator is zero. So, we set x0x \neq 0 to ensure the function is defined. The domain of the function is therefore all real numbers except zero:

A=R{0}A = \mathbb{R} \setminus \{0\}

b) Prove that for every xAx \in A, f(x)=x2f(x) = x - 2

We simplify the function f(x)f(x):

f(x)=x22xx=x(x2)xf(x) = \frac{x^2 - 2x}{x} = \frac{x(x - 2)}{x}

For x0x \neq 0, we can cancel the xx terms:

f(x)=x2f(x) = x - 2

Thus, we have proven that f(x)=x2f(x) = x - 2 for all xAx \in A.

c) Calculate limx0f(x)\lim_{x \to 0} f(x)

From part (b), we know that f(x)=x2f(x) = x - 2 for xAx \in A. Therefore, we calculate the limit as x0x \to 0:

limx0f(x)=limx0(x2)\lim_{x \to 0} f(x) = \lim_{x \to 0} (x - 2)

This is a simple linear function, and as x0x \to 0, the limit becomes:

limx0(x2)=2\lim_{x \to 0} (x - 2) = -2

Thus, limx0f(x)=2\lim_{x \to 0} f(x) = -2.


Would you like more details on any of the parts or further clarifications?

Here are 5 related questions for practice:

  1. What happens to the function if you modify it to f(x)=x23xxf(x) = \frac{x^2 - 3x}{x}?
  2. How would you approach finding the derivative of f(x)=x2f(x) = x - 2?
  3. What would the limit be if the function were f(x)=x22xf(x) = x^2 - 2x as x0x \to 0?
  4. What is the range of f(x)=x2f(x) = x - 2?
  5. How would the domain change if f(x)=x22xx2f(x) = \frac{x^2 - 2x}{x^2}?

Tip: When simplifying rational expressions, always check for domain restrictions before canceling terms.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Functions
Limits

Formulas

f(x) = (x^2 - 2x) / x
lim (x → 0) f(x)

Theorems

Limit Theorem
Simplification of Rational Expressions

Suitable Grade Level

Grades 9-11