Range of Functions

Introduction 

A relation is a set of connections between one set and another. These connections follow some conditions defined by mathematical equations. 

Functions are the relations between a set of inputs and a set of outputs. Each input, when put into a function, gives one result. Each number put into the function provides another number. So in mathematical terms, functions map the input and their outputs. The critical thing to note is that each information has one and only one output. So while every function is a relation, not every relation is a function. A function exists because there is a set of inputs and a set of corresponding outputs. So these two sets are of paramount importance in operations involving functions.

Functions can sometimes be thought of as machines with an inlet and an outlet. The input is put in through one end, the machine that is the function processes the input, and the output comes out of the other end. 

 Range of functions

When functions are considered, they need to be taken with their domain and range. The domain is the set of inputs for which the process is defined. This means that the values for the variables in the function will be taken from the domain. When the function is applied to the domain, it produces an output for each input value. This set of outputs is known as the range of functions. The following illustration can explain this.

Suppose there is a domain set {1, -2, 2, 3} and the f(x) = then the range set would be {1, 4, 9}.

 Finding the range of the function

If there is a well-defined domain of the function, then the range can be found by putting in the variable’s value from the domain. The function equation is then solved, and the answer thus obtained is a part of the range set. The following example will illustrate this.

Suppose there is a set D = {2, 3, 4}

The f(x) = 2x-1

Then the range can be found by solving the equation for x and using the set A as the values for x. 

So the range set R = {3, 5, 7}

The set of ordered pairs from the above would be = {(2,3),(3,5),(4,7)}

• Looking at the set of the ordered pairs above, it is clear that it is a function because none of the values for x are repeated. That means each value of the domain has only one corresponding answer.

Mapping the range of functions on a graph

Mapping out functions on a graph is simply finding out the ordered set of the domain and the range of functions. Once this set is found out, the ordered pairs can be plotted on a graph at the corresponding points on the x and y axes. Joining the plotted ordered pairs on the chart will give the function a precise shape as a graph. 

 Finding the range of functions from their graphical representation

The domain and the range of functions can be represented on graphs together. For example, a graph for a quadratic function with a positive coefficient will be a u-shaped or parabolic form. Its lowest point will be at 0, from where the slope will start rising again. By this, it can be determined that the values of y are either more than or equal to 0. So the range for this function is the set of all real numbers R except y<0. Whereas if it is a cubic equation, the graph is an n shaped diagonal line. In this case, the graph line passes through the x-axis, so the range for this function would be the set of all real numbers R.

Conclusion

When considering the range of functions, it is essential to note that the range is a unique set resulting from the function defined for the domain. A function is a relation between two sets where one is a set of inputs, and the other is the set of outputs. The resulting answer is the output when the inputs are put into the function. These outputs together form a range of functions. Another essential characteristic of functions is that each information has one and only one output. This is something that defines functions as relations. So even though all functions are relations, all relations are not functions.