Determine the domain of a function from an equation

The domain of a real-valued function defined by an equation is the set of all real input values for which the expression is meaningful: polynomial expressions allow all real numbers, while rational expressions exclude values making a denominator zero, even-root expressions require nonnegative radicands, and logarithmic expressions require positive arguments. The domain is represented with set-builder or interval notation and is distinguished from the range; complex-valued domains and more abstract domain conventions are outside this scope.

Detailed Explanation: Determine the domain of a function from an equation

The domain of a function is the set of all real numbers that can be used as inputs without making the equation undefined.

Check for these restrictions:

A denominator cannot equal $0$ .
An even-root radicand must be greater than or equal to $0$ .
A logarithm’s argument must be greater than $0$ .
A polynomial has domain all real numbers.

Example

Determine the domain of

f(x)=\frac{\sqrt{x+2}}{x-3}.

Step 1: Check the square root

The expression inside the square root must be nonnegative:

x+2\ge 0

Subtract $2$ :

x\ge -2.

So the square root allows inputs in the interval

[-2,\infty).

Step 2: Check the denominator

The denominator cannot be zero:

x-3\ne 0.

Therefore,

x\ne 3.

Step 3: Combine the restrictions

We need $x\ge -2$ , but we must leave out $x=3$ . Thus, the domain is

\boxed{[-2,3)\cup(3,\infty)}.

The bracket at $-2$ shows that $-2$ is included, because $\sqrt{0}$ is defined. The number $3$ is excluded because it would make the denominator equal to zero.

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Function Domain - Simple Quadratic to Domain Definition

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