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Unit 5: Exponents, Roots, and Rational Expressions

Topic 4

Domain Restrictions and Undefined Values

A domain is the set of all input values (xx) for which an expression or function is defined. When working with rational expressions, radicals, or denominators, certain values must be excluded from the domain because they make the expression undefined.
1. Division by zero is undefined: Any value that makes a denominator equal to zero must be excluded.
1x−2 is undefined at x=2.\frac{1}{x-2} \text{ is undefined at } x = 2.
2. Even roots of negative numbers are undefined for real values: For example, x−3\sqrt{x-3} is defined only when x−3≥0x - 3 \ge 0.
3. Numerators do not affect the domain, only denominators and even roots do.
The SAT often tests whether you can identify these restrictions when simplifying or interpreting rational or radical expressions.

Core Skills

  • Identify values that make denominators zero.
  • Determine where expressions under even roots are nonnegative.
  • Express domain restrictions in set notation or interval notation.
  • Retain restrictions even after simplification.
  • Check all variable conditions before evaluating or graphing.

Example 1: Denominator Restriction

Find the domain of 3x+1x−4\dfrac{3x + 1}{x - 4}.
x−4=0⇒x=4x - 4 = 0 \Rightarrow x = 4
The expression is undefined at x=4x = 4. Domain: all real numbers except 4
x≠4\boxed{x \ne 4}

Example 2: Multiple Denominators

Find the domain of x+2x2−9\dfrac{x + 2}{x^2 - 9}.
x2−9=(x−3)(x+3)x^2 - 9 = (x - 3)(x + 3)
Undefined when x=3x = 3 or x=−3x = -3. Domain: all real xx except x=±3x = \pm 3.
x≠3,−3\boxed{x \ne 3, -3}

Example 3: Even Root Restriction

Find the domain of x−5\sqrt{x - 5}.
x−5≥0⇒x≥5x - 5 \ge 0 \Rightarrow x \ge 5
Domain: [5,∞)[5, \infty)

Example 4: Radical in a Denominator

Find the domain of 1x−2\dfrac{1}{\sqrt{x - 2}}.
x−2>0⇒x>2x - 2 > 0 \Rightarrow x > 2
The denominator cannot be zero or negative. Domain: (2,∞)(2, \infty)

Example 5: Rational Expression After Simplification

Simplify and find the domain of x2−9x−3\dfrac{x^2 - 9}{x - 3}.
(x−3)(x+3)x−3=x+3\dfrac{(x - 3)(x + 3)}{x - 3} = x + 3
Even though the simplified form is x+3x + 3, the restriction x≠3x \ne 3 remains because the original denominator was zero at x=3x = 3. Final Answer: x+3,  x≠3\boxed{x + 3, \; x \ne 3}

Example 6: Combined Root and Denominator

Find the domain of x+4x−2\dfrac{\sqrt{x + 4}}{x - 2}.
x+4≥0⇒x≥−4,x−2≠0⇒x≠2x + 4 \ge 0 \Rightarrow x \ge -4, \quad x - 2 \ne 0 \Rightarrow x \ne 2
Domain: all x≥−4x \ge -4 except x=2.x = 2.
x≥−4,  x≠2\boxed{x \ge -4, \; x \ne 2}

Key Takeaways

  • Denominators cannot be zero; exclude those xx-values.
  • Even roots require nonnegative radicands (x≥0x \ge 0).
  • Simplifying does not remove restrictions from the original expression.
  • Odd roots and numerators impose no restrictions.
  • Always check both radicals and denominators together for valid domains.