Determine whether the equation represents y as a function of x. x+y^2=3

This question aims to identify whether the given equation represents a function or not.

A function is an interpretation, principle, or rule in mathematics that characterizes an association between an independent and a dependent variable. Functions are common in mathematical concepts and are required for the formulation of physical relationships in the scientific disciplines. A variable is a notion or an element whose magnitude can be expressed numerically, that is, it can be determined numerically. Variables are so named since they differ, that is, they may contain a wide range of values. A variable therefore can be defined as a quantity that can take on several different values in a given question.

Making computations with variables as if they will represent numbers enables one to deal with a wide range of problems in a single calculation. In mathematics, the concept of a variable is important. A function $y = f(x)$ typically involves two variables, $x$ and $y$, each of which speaks to the function’s reliability and contention. The term variable comes from the fact that when the argument, which is also known as the variable of capacity, changes, the reliability varies correspondingly.

Expert Answer

Given function is:

$x+y^2=3$

Re-write the function as:

$y^2=3-x$

$y=\pm\sqrt{3-x}$                      (1)

The given equation is of a parabola that opens sideways and will not be a function since the parabola will be intersected by some vertical lines. In other words, it can be observed from equation (1) that there exists more than one value of $y$ for every value of $x$ in the domain. Thus, the given equation does not represent $y$ as a function of $x$.

Geogebra export scaled

A sideways open parabola

Example

Consider the equation $y-2x=3$. Find out whether the given equation is a function or not.

Solution

First, re-write the equation as:

$y=2x+3$

According to the definition of a function, for every $x$ value, there must be a single $y$ value. For this purpose, take $x=-1,0,3$ to check whether the given equation is a function or not.

At $x=-1$:

$y=2(-1)+3=1$

At $x=0$:

$y=2(0)+3=3$

At $x=3$:

$y=2(3)+3=9$

Secondly, to have sufficient reasons, observe that in the above equation, the multiplication of any $x$ value with $2$ yields a single value. Also, when $3$ is added after the multiplication, the value of $y$ remains single. Thus, the given equation represents a function.

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