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.
