Domain of a Function Defined by an Equation

The domain of a function refers to the set of all possible input values \( x \) that will produce a real output in the function. To find the domain of a function defined by an equation, we need to check for any restrictions like division by zero, square roots of negative numbers, or logarithms of non-positive numbers.

When a function is defined by an equation, the domain includes all 𝑥-values for which the equation produces a real number as output.

Key Steps for Finding the Domain

  1. Check for division by zero: Any 𝑥-value that makes the denominator zero must be excluded from the domain.
  2. Check for square roots: The expression inside the square root (or any even root) must be non-negative, as the square root of a negative number is undefined in real numbers.
  3. Check for logarithms: The argument of a logarithm must be positive.

Examples

Example 1: Linear Function

Consider the linear function:

$$ f(x) = 3x + 5 $$

There are no restrictions, so the domain is:

$$ \text{Domain: } (-\infty, \infty) $$

Example 2: Rational Function

For the function:

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

The denominator must not be zero, so \( x = 1 \) is excluded. The domain is:

$$ \text{Domain: } (-\infty, 1) \cup (1, \infty) $$

Example 3: Square Root Function

For the square root function:

$$ f(x) = \sqrt{x - 4} $$

The expression under the square root must be non-negative, so the domain is:

$$ \text{Domain: } [4, \infty) $$

Example 4: Logarithmic Function

For the logarithmic function:

$$ f(x) = \log(x - 2) $$

The argument of the logarithm must be positive, so the domain is:

$$ \text{Domain: } (2, \infty) $$

Example 5: Rational Function with Square Root

For the function:

$$ f(x) = \frac{\sqrt{x - 1}}{x + 2} $$

We need to check both the square root and the denominator:

  1. The expression inside the square root, 𝑥 − 1, must be non-negative, that is, $$ x−1≥0⟹x≥1 $$
  2. The denominator 𝑥 + 2 must not be zero, that is, $$ x + 2 \neq 0⟹x \neq − 2 $$

Thus, the domain excludes 𝑥 = − 2 and requires 𝑥 ≥ 1. Therefore, the domain is:

$$ \text{Domain: } [1, \infty) $$

Finding the domain of a function defined by an equation