Question

How do you decide whether the relation \[x={{y}^{2}}\] defines a function?

Answer 1

Hint: Draw the graph of the parabola \[x={{y}^{2}}\]. Now, draw vertical lines which are parallel to the y – axis passing through the parabola. If any one of these vertical lines passes through two points on the parabola which have different y – coordinates then the given relation will not be a function.

Complete step by step answer:
Here, we have been provided with the relation \[x={{y}^{2}}\] and we are asked to check if it is a function or not.
Now, a relation can only be a function if for no value of x there are more than one value of y. To check if a relation is a function or not we have many methods but the graphical method is one of the easiest methods. In this method first draw the graph of the given relation, then we draw vertical lines parallel to y – axis. If any one of these vertical lines parallel to y – axis. If any one of these vertical lines cuts the graph at more than one point then the relation is not considered as a function.
Now, let us come to the question. We have the relation \[x={{y}^{2}}\]. Clearly, we can see that theis equation denotes a parabola. So, let us draw this parabola on the graph.

Let us draw some vertical lines on the graph of the above parabola. So, it will look like,

As we can see that there are many lines which are cutting the parabola at more than one point, that is they are cutting the parabola at two points. So, for a single value of x there are two values of y.

Hence, we can conclude that the relation \[x={{y}^{2}}\] is not a function.

Note: One may note that if we have a relation in which we have one value of y for more than one value of x then the relation will be considered as a function. That is called a many – one function. You must remember the vertical line test that we have used to solve the question. Remember the basic differences between a ‘relation’ and a ‘function’ to solve the above question.