How to determine the domain and codomain of a given function?

Determining the Domain and Codomain of a Function

In mathematics, the domain of a function is the set of all possible inputs (or independent variables) that can be used to generate outputs (or dependent variables) with the function. The codomain, on the other hand, is the set of all possible outputs that the function can produce.

To determine the domain and codomain of a given function, follow these steps:

1. Identify the function's expression:

• The function's expression is the equation that defines the relationship between the input (x) and output (f(x)).

2. Find the domain:

• The domain is the set of all possible values that x can take.
• To find the domain, you need to determine which values of x make the function's expression undefined or produce an error, such as division by zero, square root of a negative number, or logarithm of a non-positive number.
• Exclude these values from the domain.
• The domain can be represented using interval notation (e.g., [a, b], (a, b), [a, b)), listing values (e.g., {1, 2, 3}), or using other set notation.

3. Determine the codomain:

• The codomain is the set of all possible values that f(x) can take.
• To find the codomain, consider the function's expression and the domain you've determined.
• As you evaluate the function for each value in the domain, note the range of possible outputs.
• The codomain can be represented using interval notation, listing values, or using other set notation.

Example: Consider the function f(x) = √(x - 1).

• Domain: To find the domain, we need to ensure the expression inside the square root is non-negative. So, we solve the inequality x - 1 ≥ 0, which gives us x ≥ 1. Therefore, the domain is [1, ∞).
• Codomain: As x increases from 1 to ∞, the output of f(x) also increases from 0 to ∞. Therefore, the codomain is [0, ∞).