Finding a Function's Domain

If a function $f$ does not model data or verbal conditions, its domain is the largest set of real numbers for which the value of $f$ is a real number.

Key Restrictions on Domain

Exclude real numbers that cause division by zero.
Exclude real numbers that result in an even root (e.g., square root) of a negative number.

Examples

Example A

Find the domain of $f(x) = x^2 + 3x - 17$ .

There is no denominator and no even root.
Therefore, the domain of $f$ is all real numbers, or $(-\infty, \infty)$ .

Example B

Find the domain of $g(x) = \frac{5x}{x^2 - 49}$ .

We have a denominator, so we need to find values of $x$ that make the denominator zero.
Set $x^2 - 49 = 0$ to find the restricted values.
Factor: $(x + 7)(x - 7) = 0$ .
So, $x = -7$ or $x = 7$ .
Therefore, the domain is all real numbers except $-7$ and $7$ .
In interval notation: $(-\infty, -7) \cup (-7, 7) \cup (7, \infty)$ .

Example C

Find the domain of $h(x) = \sqrt{9x - 27}$ .

We have an even root, so the expression inside the square root must be greater than or equal to zero.
Solve $9x - 27 \geq 0$ to find the restricted values.
Add 27 to both sides: $9x \geq 27$ .
Divide by 9: $x \geq 3$ .
Therefore, the domain of $h$ is all real numbers greater than or equal to 3.
In interval notation: $[3, \infty)$ .

Example D

Find the domain of $j(x) = \frac{5x}{\sqrt{24 - 3x}}$ .

We have a denominator and an even root.
The expression inside the square root must be positive (not zero, because it's in the denominator).
Solve $24 - 3x > 0$ .
Subtract 24 from both sides: $-3x > -24$ .
Divide by -3 (and reverse the inequality): $x < 8$ .
Therefore, the domain of $j$ is all real numbers less than 8.
In interval notation: $(-\infty, 8)$ .

Your Turn

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

We need to exclude values that make the denominator zero.
Set $x^2 - 4x = 0$ .
Factor: $x(x - 4) = 0$ .
So, $x = 0$ or $x = 4$ .
Therefore, the domain is $(-\infty, 0) \cup (0, 4) \cup (4, \infty)$ .