2.3 Domain and Range

The

domain

of a function is the set of all possible input values (usually \(x\)) for which the function is defined. The

range

is the set of all possible output values (usually \(y\)) the function can produce.

Key Concepts:

• Rational Expressions:

  Denominators must not be zero. Exclude values of \(x\) that make the denominator zero. \[ f(x) = \frac{1}{x - 3} \quad \text{Domain: } x \neq 3 \]

• Irrational Expressions:

  Even roots must have non-negative radicands (the expression inside the root). \[ f(x) = \sqrt{x - 2} \quad \text{Domain: } x \geq 2 \]

• Logarithmic Functions:

  The argument of the logarithm must be positive. \[ f(x) = \log(x - 1) \quad \text{Domain: } x > 1 \]

• Undefined Expressions:

  These occur when mathematical operations are not valid for certain input values. Common cases include:
  • Division by zero: \[ \frac{1}{x - 2} \quad \text{is undefined at } x = 2 \]
  • Even roots of negative numbers (in real numbers): \[ \sqrt{x - 5} \quad \text{is undefined for } x < 5 \]
  • Logarithm of non-positive numbers: \[ \log(x) \quad \text{is undefined for } x \leq 0 \]
  To find the domain, exclude values of \(x\) that make the expression undefined.

Examples:

Example 1: Domain of a Rational Function

Find the domain of \( f(x) = \frac{2x + 1}{x^2 - 4} \).

Solution:

Step 1: Set the denominator not equal to zero: \[ x^2 - 4 \neq 0 \Rightarrow x \neq \pm2 \] Therefore, the domain is all real numbers except \( x = -2 \) and \( x = 2 \).

Example 2: Domain of a Root Function

Find the domain of \( f(x) = \sqrt{5 - x} \).

Solution:

The expression under the square root must be \(\geq 0\): \[ 5 - x \geq 0 \Rightarrow x \leq 5 \] So, the domain is \( x \leq 5 \).

Example 3: Domain of a Logarithmic Function

Find the domain of \( f(x) = \log(x^2 - 9) \).

Solution:

The argument of the log must be positive: \[ x^2 - 9 > 0 \Rightarrow (x - 3)(x + 3) > 0 \] This inequality is satisfied when \( x < -3 \) or \( x > 3 \). So the domain is \( (-\infty, -3) \cup (3, \infty) \).

Example 4: Undefined Value in a Rational Expression

Find the value of \(x\) for which the expression \[ f(x) = \frac{x + 5}{x + 7} \] is undefined.

Solution:

A rational expression is undefined when the denominator is zero. Set the denominator equal to zero: \[ x + 7 = 0 \Rightarrow x = -7 \] Therefore, the expression is undefined at \( \boxed{x = -7} \).

Example 5: Undefined Values in a Rational Expression

Find the values of \(x\) for which the expression \[ f(x) = \frac{1}{(x - 3)(x^2 - 9)} \] is undefined.

Solution:

Step 1: Set the denominator equal to zero: \[ (x - 3)(x^2 - 9) = 0 \] Step 2: Factor further: \[ x^2 - 9 = (x - 3)(x + 3) \] So the full denominator becomes: \[ (x - 3)(x - 3)(x + 3) \] Step 3: Set each factor equal to zero: \[ x - 3 = 0 \Rightarrow x = 3 \] \[ x + 3 = 0 \Rightarrow x = -3 \] Therefore, the expression is undefined at \( \boxed{x = -3 \text{ and } x = 3} \).