Problem 140 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the equation represents \(y\) as a function of \(x .\) $$x-y^{2}=0$$

Step-by-Step Solution

Answer

The equation \(x - y^{2} = 0\) does not represent \(y\) as a function of \(x\), as there is more than one possible \(y\) value for each positive \(x\) value.

Step 1 : Solve for y

Rearrange the given equation to solve for \(y\). This means isolating \(y\) on one side. Doing this we get the equation \(y^{2} = x\).

Step 2 : Find the square root of each side

To fully isolate \(y\), we'll have to take the square root of the each side of the equation. This operation results in \(y = ± \sqrt{x}\). The ± sign indicates that there are two values for \(y\) for each positive \(x\) value. Negative \(x\) values cannot be solved for \(y\) because the square root of a negative number is not a real number.

Step 3 : Conclusion

Given the equation \(y = ± \sqrt{x}\), it's evident that there is not a single unique \(y\) for each \(x\) value. Thus, the initial equation does not represent \(y\) as a function of \(x\).

Key Concepts

Solving Quadratic Equations

Understanding how to solve quadratic equations is one of the fundamental skills in algebra. A quadratic equation generally takes the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants with \(a \eq 0\). The solutions to these equations are known as the 'roots' and can be found using various methods, including factoring, completing the square, or using the quadratic formula.

In the context of the exercise provided, the equation is already in an easily solvable form, \(y^2 = x\). The goal is to isolate the variable \(y\) by performing the inverse operation of squaring, which is taking the square root. This process leads to two potential solutions, positive and negative, because squaring either a positive or negative number gives a positive result. These roots are critical in understanding the nature of the solutions to quadratic equations and in determining whether the equation represents a function.

Square Root Functions

Square root functions are a type of radical functions. The basic form of a square root function is \(y = \sqrt{x}\), where \(y\) is expressed in terms of the square root of \(x\). This type of function usually involves half of a parabola that opens upward and starts from the origin if there are no additional transformations.

When solving an equation like \(y^2 = x\), taking the square root of both sides, as indicated in the solution, introduces the \(\pm\) symbol, signifying that there are two possible values of \(y\) for each positive \(x\): \(y = \sqrt{x}\) or \(y = -\sqrt{x}\). However, it is important to note that for a relationship to be considered a function, each input \(x\) should correspond to exactly one output \(y\). The presence of the \(\pm\) suggests multiple outputs for each input, which takes us to our next point about function uniqueness.

Function Uniqueness in Mathematics

Function uniqueness is a principle that holds a special place in mathematics, particularly in defining what a function is. A function is a relationship between inputs and outputs where each input is associated with exactly one unique output. This concept is known as the Vertical Line Test in graphing: if a vertical line intersects a graph at more than one point, then the graph does not represent a function.

Returning to our exercise, the equation \(y = \pm \sqrt{x}\) fails to meet this criterion of uniqueness, as for every positive input \(x\), there are two different outputs \(y\). Thus, this rule is crucial when determining whether a mathematical relationship qualifies as a function, and it's why we conclude that the given equation does not define \(y\) as a function of \(x\). The understanding of function uniqueness is paramount when analyzing equations and their graphs to ascertain the function status.