The domain of a function is the set of all possible input values (usually x-values) for which the function is defined. Finding the domain involves identifying the values that x can take without causing any mathematical contradictions or undefined scenarios (like division by zero or square roots of negative numbers).
Steps to Find the Domain of a Function
- Look for any fractions in the function. The denominator cannot be zero, so exclude values of x that make the denominator zero.
- Check for square roots or other even roots. The radicand (the number under the root) must be non-negative, so determine the x-values that satisfy this condition.
- Consider the context of the problem. Sometimes, the domain is restricted by the real-world situation being modeled by the function.
Example 1: Polynomial Function
Find the domain of f(x) = x^2 + 3x – 4.
This is a polynomial function, and polynomials are defined for all real numbers. Therefore, the domain of f(x) is all real numbers.
Example 2: Rational Function
Find the domain of f(x) = 1 / (x – 2).
Since the denominator cannot be zero, x cannot be 2. Therefore, the domain of f(x) is all real numbers except x = 2.
Example 3: Function with a Square Root
Find the domain of f(x) = √(5 – x).
The expression under the square root, 5 – x, must be non-negative. So, 5 – x ≥ 0, which means x ≤ 5. Therefore, the domain of f(x) is x ≤ 5.
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