For What Values Of X Is The Expression Below Defined

Determining the Domain: For What Values of x is the Expression Defined?

Understanding the domain of a mathematical expression is a foundational skill in algebra and calculus. It answers the critical question: "Which input values (x) can I safely plug into this expression without causing a mathematical error?" The expression we will analyze is:

√(x² - 4) / (x² - 4x + 3)

To find all values of x for which this expression is defined in the real number system, we must identify and combine the restrictions imposed by two key components: the square root in the numerator and the polynomial denominator. An expression is undefined if it involves taking the square root of a negative number or dividing by zero. Our task is to systematically eliminate any x-values that lead to these scenarios.

Step-by-Step Domain Analysis

We will analyze the numerator and denominator separately, then find the intersection of their allowed values.

1. Restriction from the Square Root (Numerator)

The expression inside the square root, called the radicand, must be greater than or equal to zero for the result to be a real number.

• Condition: x² - 4 ≥ 0
• Solve the Inequality: This factors neatly as a difference of squares: (x - 2)(x + 2) ≥ 0.
• Critical Points: The roots are x = 2 and x = -2. These points divide the number line into three intervals: (-∞, -2), (-2, 2), and (2, ∞).
• Test Intervals:
  • For x < -2 (e.g., x = -3): (-3-2)(-3+2) = (-5)(-1) = +5 ≥ 0 → Allowed.
  • For -2 < x < 2 (e.g., x = 0): (0-2)(0+2) = (-2)(2) = -4 < 0 → Not Allowed.
  • For x > 2 (e.g., x = 3): (3-2)(3+2) = (1)(5) = +5 ≥ 0 → Allowed.
• Include the Critical Points: At x = -2 and x = 2, the radicand is 0, and √0 = 0, which is perfectly defined. Therefore, these endpoints are included.
• Result from Numerator: x ∈ (-∞, -2] ∪ [2, ∞). This means x must be less than or equal to -2 OR greater than or equal to 2.

2. Restriction from the Denominator

A fraction is undefined when its denominator equals zero.

• Condition: x² - 4x + 3 ≠ 0
• Solve the Equation: Factor the quadratic: (x - 1)(x - 3) ≠ 0.
• Find the Excluded Values: Set each factor to zero: x - 1 = 0 → x = 1; x - 3 = 0 → x = 3.
• Result from Denominator: x ≠ 1 and x ≠ 3.

3. Combining the Restrictions (The Final Domain)

The expression is only defined where BOTH conditions are satisfied simultaneously. We start with the set from the numerator and then remove the values excluded by the denominator.

1. Our candidate set from the square root is: (-∞, -2] ∪ [2, ∞).
2. Now, remove the points x = 1 and x = 3 from this candidate set.
   • Is x = 1 in our candidate set? 1 is not in (-∞, -2] and not in [2, ∞). Therefore, x = 1 is already excluded by the square root's requirement. We don't need to worry about it further.
   • Is x = 3 in our candidate set? Yes, 3 is in the interval [2, ∞). Therefore, we must explicitly exclude x = 3.

Final Domain: The set of all real numbers x such that x ≤ -2 OR (x ≥ 2 AND x ≠ 3). In interval notation, this is written as: (-∞, -2] ∪ [2, 3) ∪ (3, ∞)

Scientific Explanation: Why These Rules Exist

The restrictions are not arbitrary; they are fundamental properties of the real number system.

• The Square Root Function (√): By definition, the principal square root function f(y) = √y is only defined for y ≥ 0 within the real numbers (ℝ). There is no real number that, when squared, yields a negative result. Attempting to compute √(-1) in the real number system leads to an imaginary number (i), which is outside our domain of consideration unless specified. Therefore, the radicand must be non-negative.
• Division by Zero: The operation of division is defined as multiplication by the multiplicative inverse. For any real number a, its inverse is 1/a. However, zero has no multiplicative inverse. There is no real number that, when multiplied by 0, gives 1 (since 0 × anything = 0). Therefore, any expression of the form something / 0 is undefined. This is a hard, non-negotiable rule in arithmetic.

Our expression is a composite function of these two operations. For the entire expression to yield a single, well-defined real number, both the numerator's operation (square root) and the denominator's operation (division) must be valid independently. The domain is the intersection of their individual valid input sets.

Common Pitfalls and FAQs

Q1: I solved x² - 4 ≥ 0 and got x ≥ 2 or x ≤ -2, but I forgot about the denominator. Is my answer complete? A: No. A complete domain requires checking