Chapter 2.1, Problem 27AYU

To determine

Whether the equation $y=x^2$ is a function of x .

Answer to Problem 27AYU

The equation $y=x^2$ defines y as a function of x .

Explanation of Solution

Consider two sets A and B . If each element $x$ in the set A has exactly one element $y$ in the set B , the relation between them is known as a function.The ordered pair is $\left(x,y\right)$ .

The values of first coordinate $x$ is known as input values and the values of second coordinate $y$ is known as output values of a function.

Therefore, the set of input values represent the domain and set of output values represent the range of the function.

If one or more elements in the set A has more than one image in the set B , then the relation will not be a function.

So, every function is a relation but every relation is not a function.

Now, consider the equation.

  $y=x^2$

Now, let us take some values of x as $1,2,3$ .

Substitute the values in the equation.

  $\begin{array}{l}y={\left(1\right)}^2=1\\ y={\left(2\right)}^2=2\\ y={\left(3\right)}^2=3\end{array}$

Therefore, the equation gives exactly one value of y for each x .

Here x is independent variable and y is dependent variable.

Hence, the equation $y=x^2$ defines y as a function of x .