Math Problem Statement

Solve

Solution

To find the domain of each rational function, we must ensure that the denominator is not equal to zero, as dividing by zero is undefined. Let's go through each function step by step:

a) f(x)=2xf(x) = \frac{2}{x}

  • Denominator: xx
  • Set the denominator xx ≠ 0: x0x \neq 0
  • Domain: All real numbers except x=0x = 0. Domain: (,0)(0,)\text{Domain: } (-\infty, 0) \cup (0, \infty)

b) f(x)=1x+2f(x) = \frac{1}{x + 2}

  • Denominator: x+2x + 2
  • Set the denominator x+2x + 2 ≠ 0: x+20x2x + 2 \neq 0 \quad \Rightarrow \quad x \neq -2
  • Domain: All real numbers except x=2x = -2. Domain: (,2)(2,)\text{Domain: } (-\infty, -2) \cup (-2, \infty)

c) f(x)=1x+1f(x) = \frac{1}{x} + 1

  • Denominator: xx
  • Set the denominator xx ≠ 0: x0x \neq 0
  • Domain: All real numbers except x=0x = 0. Domain: (,0)(0,)\text{Domain: } (-\infty, 0) \cup (0, \infty)

d) f(x)=2x2f(x) = \frac{-2}{x^2}

  • Denominator: x2x^2
  • Set the denominator x2x^2 ≠ 0: x20x0x^2 \neq 0 \quad \Rightarrow \quad x \neq 0
  • Domain: All real numbers except x=0x = 0. Domain: (,0)(0,)\text{Domain: } (-\infty, 0) \cup (0, \infty)

e) f(x)=1(x3)2f(x) = \frac{1}{(x - 3)^2}

  • Denominator: (x3)2(x - 3)^2
  • Set the denominator (x3)2(x - 3)^2 ≠ 0: (x3)20x3(x - 3)^2 \neq 0 \quad \Rightarrow \quad x \neq 3
  • Domain: All real numbers except x=3x = 3. Domain: (,3)(3,)\text{Domain: } (-\infty, 3) \cup (3, \infty)

f) f(x)=1(x+2)23f(x) = \frac{-1}{(x + 2)^2} - 3

  • Denominator: (x+2)2(x + 2)^2
  • Set the denominator (x+2)2(x + 2)^2 ≠ 0: (x+2)20x2(x + 2)^2 \neq 0 \quad \Rightarrow \quad x \neq -2
  • Domain: All real numbers except x=2x = -2. Domain: (,2)(2,)\text{Domain: } (-\infty, -2) \cup (-2, \infty)

Summary of Domains:

  • a) (,0)(0,)(-\infty, 0) \cup (0, \infty)
  • b) (,2)(2,)(-\infty, -2) \cup (-2, \infty)
  • c) (,0)(0,)(-\infty, 0) \cup (0, \infty)
  • d) (,0)(0,)(-\infty, 0) \cup (0, \infty)
  • e) (,3)(3,)(-\infty, 3) \cup (3, \infty)
  • f) (,2)(2,)(-\infty, -2) \cup (-2, \infty)

Do you need further details or have any questions?

Here are five related questions for further practice:

  1. How would the domain change if the denominator of a rational function was a cubic polynomial?
  2. What is the domain of the function g(x)=5x24g(x) = \frac{5}{x^2 - 4}?
  3. Can the domain of a rational function ever be all real numbers? Why or why not?
  4. How do we determine the domain of a rational function if the denominator is a quadratic expression?
  5. What are the steps to finding the domain of a rational function with multiple factors in the denominator?

Tip: Always factorize the denominator when possible to easily identify restrictions on the domain.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Domain of Functions

Formulas

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Theorems

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Suitable Grade Level

Grades 10-12