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How to Find the Domain of a Function with Division?

Published in Function Domains 3 mins read

The domain of a function involving division is found by identifying all real numbers that, when inputted, do not result in division by zero.

Here's a breakdown of the process:

Understanding the Issue

Division by zero is undefined in mathematics. Therefore, when a function includes a fraction (or rational expression), we must exclude any values of the variable that make the denominator equal to zero.

Steps to Find the Domain

  1. Identify the Denominator: Locate the denominator of the fractional part of the function.

  2. Set the Denominator Equal to Zero: Write an equation where the denominator is equal to zero.

  3. Solve for the Variable: Solve the equation you created in step 2. The solutions represent the values that must be excluded from the domain.

  4. Express the Domain: Write the domain in one of the following ways:

    • Set Notation: {x | x ≠ a, x ≠ b, ...}, where a, b, etc., are the values you found in step 3. This reads, "the set of all x such that x is not equal to a, x is not equal to b, and so on."
    • Interval Notation: (-∞, a) ∪ (a, b) ∪ (b, ∞), etc., where a, b, etc., are the values you found in step 3, listed in increasing order. The "∪" symbol means "union," indicating that the domain includes all of these intervals.

Example

Let's say you have the function:

f(x) = 1 / (x - 3)

  1. Denominator: The denominator is (x - 3).

  2. Set to Zero: x - 3 = 0

  3. Solve: x = 3

  4. Express the Domain:

    • Set Notation: {x | x ≠ 3}
    • Interval Notation: (-∞, 3) ∪ (3, ∞)

This means the function is defined for all real numbers except for x = 3.

More Complex Examples

  • f(x) = (x + 2) / (x2 - 4)

    1. Denominator: x2 - 4
    2. Set to Zero: x2 - 4 = 0
    3. Solve: (x + 2)(x - 2) = 0 => x = -2 or x = 2
    4. Domain: {x | x ≠ -2, x ≠ 2} or (-∞, -2) ∪ (-2, 2) ∪ (2, ∞)

  • f(x) = x / (x2 + 1)

    1. Denominator: x2 + 1
    2. Set to Zero: x2 + 1 = 0
    3. Solve: x2 = -1 => No real solutions (because the square of a real number cannot be negative).
    4. Domain: All real numbers, which can be written as {x | x ∈ ℝ} or (-∞, ∞)

Key Considerations

  • Other Restrictions: Remember that functions can have domain restrictions for reasons other than division by zero (e.g., square roots of negative numbers, logarithms of non-positive numbers). You must consider all potential restrictions when determining the overall domain.
  • Intersection of Domains: If your function involves multiple parts with domain restrictions (e.g., a fraction and a square root), the overall domain is the intersection of the individual domains. This is described in the reference as A ∩ B where A and B are the domains of the functions f(x) and g(x), respectively.

In summary, finding the domain of a function with division involves identifying values that would make the denominator zero and excluding those values from the set of all real numbers.

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