B. Domain and Range

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 of a Function

A domain of a function refers to "all the values" that go into a function. The domain of a function is the set of all possible inputs for the function. Consider this box as a function f(x) = 2x . Inputting the values x = {1,2,3,4,...}, the domain is simply the set of natural numbers and the output values are called the range. But in general, f(x) = 2x is defined for all real values of x and hence its domain is the set of all real numbers which is denoted by (-∞, ∞). Here are the general formulas used to find the domain of different types of functions. Here, R is the set of all real numbers.

· Domain of any polynomial (linear, quadratic, cubic, etc) function is R.

· Domain of a square root function √x is x≥0.

· Domain of an exponential function is R.

· Domain of logarithmic function is x>0.

· To find the domain of a rational function y = f(x), set the denominator ≠ 0.

Range of a Function

The range of a function is the set of all its outputs. Example: Let us consider the function f: A→ B, where f(x) = 2x and each of A and B = {set of natural numbers}. Here we say A is the domain and B is the co-domain. Then the output of this function becomes the range. The range = {set of even natural numbers}. The elements of the domain are called pre-images and the elements of the co-domain which are mapped are called the images. Here, the range of the function f is the set of all images of the elements of the domain (or) the set of all the outputs of the function. In the upcoming sections, we can see how to find the range of different types of functions. Here are the general formulas used to find the range of different types of functions. Note that R is the set of all real numbers here.

· Range of a linear function is R.

· Range of a quadratic function y = a(x-h)² + k is:

y≥k, if a>0 and

y≤k, if a<0

· Range of a square root function is y≥0.

· Range of an exponential function is y>0.

· Range of logarithmic function is R.

· To find the range of a rational function y = f(x), solve it for x and set the denominator ≠ 0.