**How to Find the Domain of a Function**

The domain of a function is the set of all possible input values for which the function is defined. In other words, it is the set of all x-values for which the function has a corresponding y-value.

To find the domain of a function, you need to look for any restrictions on the input values. These restrictions can be caused by:

**Division by zero:**The domain of a function cannot include any values that would cause the denominator of a fraction to be zero.**Square roots:**The domain of a function cannot include any values that would cause the radicand of a square root to be negative.**Logarithms:**The domain of a function cannot include any values that would cause the argument of a logarithm to be zero or negative.**Rational exponents:**The domain of a function cannot include any values that would cause the base of a rational exponent to be zero or negative.

**Example 1:** Find the domain of the function f(x) = 1/(x-2).

Solution: The function f(x) has a restriction on the input values because of the denominator of the fraction. The denominator cannot be zero, so x-2 cannot be zero. Therefore, the domain of f(x) is all real numbers except for x = 2.

**Example 2:** Find the domain of the function g(x) = √(x+3).

Solution: The function g(x) has a restriction on the input values because of the radicand of the square root. The radicand cannot be negative, so x+3 must be greater than or equal to zero. Therefore, the domain of g(x) is all real numbers greater than or equal to -3.

**Example 3:** Find the domain of the function h(x) = log(x-1).

Solution: The function h(x) has a restriction on the input values because of the argument of the logarithm. The argument of the logarithm cannot be zero or negative, so x-1 must be greater than zero. Therefore, the domain of h(x) is all real numbers greater than 1.

**Example 4:** Find the domain of the function j(x) = x^(1/3) + 2.

Solution: The function j(x) has a restriction on the input values because of the rational exponent. The base of the rational exponent cannot be zero or negative, so x must be greater than zero. Therefore, the domain of j(x) is all real numbers greater than zero.

**FAQ**

**What is the difference between the domain and the range of a function?**

The domain of a function is the set of all possible input values, while the range of a function is the set of all possible output values.

**Can a function have more than one domain?**

No, a function can only have one domain. However, a function can have more than one range.

**What is the domain of a constant function?**

The domain of a constant function is the set of all real numbers.

**What is the domain of a linear function?**

The domain of a linear function is the set of all real numbers.

**What is the domain of a quadratic function?**

The domain of a quadratic function is the set of all real numbers.

**What is the domain of a cubic function?**

The domain of a cubic function is the set of all real numbers.

**What is the domain of a rational function?**

The domain of a rational function is the set of all real numbers except for the values that would cause the denominator of the fraction to be zero.

**What is the domain of a radical function?**

The domain of a radical function is the set of all real numbers that make the radicand non-negative.

**What is the domain of a logarithmic function?**

The domain of a logarithmic function is the set of all positive real numbers.

**What is the domain of an exponential function?**

The domain of an exponential function is the set of all real numbers.