Answer:
The domain and range are important concepts in mathematics that are used to describe the input and output of a function or relation, respectively.
1. Domain:
The domain of a function or relation refers to the set of all possible input values or independent variables. It represents the values that can be plugged into the function or relation to generate an output.
For example, consider the function f(x) = x^2. In this case, the domain would be all real numbers because any real number can be squared to produce a valid output. So, the domain of f(x) = x^2 is (-∞, ∞), which means that any real number can be an input for this function.
However, it's important to note that certain functions may have restrictions on their domain. For instance, a function like f(x) = 1/x cannot accept the value x = 0 because division by zero is undefined. Therefore, the domain of f(x) = 1/x would be all real numbers except x = 0, which can be represented as (-∞, 0) U (0, ∞).
2. Range:
The range of a function or relation refers to the set of all possible output values or dependent variables. It represents the values that the function or relation can produce based on its given inputs.
Continuing with the previous example of f(x) = x^2, the range would be all non-negative real numbers, including zero. This is because squaring any real number always produces a non-negative result.
Similar to the domain, the range of a function can also be restricted. For instance, a function like f(x) = sin(x) has a range of [-1, 1] because the sine function oscillates between -1 and 1.
It's worth noting that the domain and range of a function can vary depending on the specific context or restrictions imposed by the function. Therefore, it is essential to carefully consider the given function and its properties to determine the accurate domain and range.