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What is a domain in math

# What is a domain in math

It’s actually hard to define. A domain can be either a set of variables, numbers, or functions that are related. Sometimes they are also called restriction, exterior, or open sets.The domain and range of a function are the components of a function. The domain is the set of all the input values of a function and range is the possible output given by the function. Domain→ Function →Range. If there exists a function f: A →B such that every element of A is mapped to elements in B, then A is the domain and B is the co-domain. The image of an element 'a' under a relation R is given by 'b', where (a,b) ∈ R. The range of the function is the set of images. The domain and range of a function is denoted in general as follows: Domain(f) = {x ∈ R} and range(f)={f(x) : x ∈ domain(f)}

## Domain

via GIPHY

Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an even root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.

Yes. For example, the function $$f(x)=-\dfrac{1}{\sqrt{x}}$$ has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function’s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common. (Source:math.libretexts.org)

## Set

The domain of a function is the set of input values, $x$, for which a function is defined. The domain is shown in the left oval in the picture below. The function provides an output value, $f(x)$, for each member of the domain. The set of values the function outputs is termed the range of the function, and those values are shown in the right hand oval in the picture below. A function is the relation that takes the inputs of the domain and output the values in the range. The rule for a function is that for each input there is exactly one output.

Another way to put it is that the domain of a function is the set of all possible values that x can be equal to that will make a valid equation. There are only two instances in which an equation will not be valid: if there's a zero in the denominator or a negative square root. In all other instances, the equation works. Take, for example, the function f(x) = x2.the domain and range of a relation is found as follows. Let R be the relation from a non-empty set A to a non-empty set B. The domain and range of the relation are the set of first elements and the second elements respectively in the ordered pairs in relation R is called the domain. (Source:www.cuemath.com)

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