Problem 102 - Algebra and Trigonometry

Question

Does the equation \(3 x+y^{2}=10\) define \(y\) as a function of \(x ?\) (Section \(2.1,\) Example 3 )

Step-by-Step Solution

Answer

No, the equation \(3x + y^2 = 10\) does not define \(y\) as a function of \(x\) because there seems to be two possible values of \(y\) for a single value of \(x\).

Step 1: Transform the equation in terms of \(y\)

The given equation is: \(3x + y^2 = 10\). Trying to solve for \(y\) we get \(y^2 = 10 - 3x\). Taking the square root on both sides gives \(y = \pm \sqrt{10 - 3x}\). Remember, whenever we take the square root of a number, it can either be positive or negative.

Step 2: Check for uniqueness of output

The solution in Step 1 leads to two possible values for \(y\) for a given \(x\), the positive value and the negative value. This means, for a single input \(x\), there are two corresponding outputs of \(y\).

Step 3: Final decision

Since there are two values of \(y\) for a certain value of \(x\), the equation \(3x + y^2 = 10\) does not define \(y\) as a function of \(x\). This is because a function stipulates that each input value should correspond to exactly one output value.

Key Concepts

Function Definition

A function, in the realm of algebra, is a special relationship between two sets of numbers, where each input from the first set (often called the domain) is connected to exactly one output in the second set (often called the range). This concept is crucial to understand algebraic equations and how they act under various operations.

In the context of \(y\) as a function of \(x\), for example \(y=f(x)\), the idea is that for every \(x\) value in the domain, there is precisely one \(y\) value. If this condition is fulfilled, the relationship defined by the equation is deemed a function. When a given \(x\) results in more than one possible \(y\), the equation does not represent a function.

This concept is pivotal, as it underpins many of the principles applied throughout algebra, from graphing equations to solving complex algebraic expressions.

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra that involves finding the values of \(x\) that make a quadratic polynomial equal to zero. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). To solve for \(x\), various methods can be applied such as factoring, using the quadratic formula, completing the square, or graphing.

For example, an equation of the form \(y^2 = 10 - 3x\) can be viewed as a quadratic equation in \(y\), where \(y\) can be solved in terms of \(x\). When dealing with square roots during the solving process, it's important to consider both the positive and negative square roots, as they may both represent possible solutions to the equation.

Square Roots in Algebra

In algebra, the square root operation is fundamental to solving equations involving squares. The square root of a number \(a\) is a value \(b\) that, when multiplied by itself, gives \(a\): \(b^2 = a\). It is essential to remember that most numbers have two square roots: a positive and a negative root, denoted as \( +\sqrt{a} \) and \( -\sqrt{a} \).

This concept is vital when working with quadratic equations, as the square of a variable can yield a positive or negative root. In the context of the original exercise, this is why the equation \(y^2 = 10 - 3x\) leads to a plus-minus solution \(y = \pm\sqrt{10 - 3x}\). Understanding the nature and properties of square roots is crucial for correctly solving algebraic problems and for recognizing when an equation may not define a function.

Function Uniqueness

The uniqueness of a function is an attribute that dictates that for every input, there can only be one specific output. This is often referred to as the 'Vertical Line Test' in graphing, where if a vertical line intersects the graph of an equation at more than one point, the equation cannot represent a function.

In the step-by-step solution provided, the equation \(3x + y^2 = 10\) yields two solutions for \(y\) when solved for a given \(x\). This violation of the one-to-one relationship required by a function means that not all algebraic equations with \(x\) and \(y\) represent functions of \(x\). Recognizing equation structures that fail the uniqueness criterion is as important as solving the equations themselves, as it helps in appropriately categorizing mathematical relationships.