Problem 6 - Precalculus: Mathematics for Calculus

Question

Each equation, is \(y\) directly proportional, inversely proportional, or not proportional to \(x ?\). (a) \(y=\frac{3}{x+1}\) (b) \(y=\frac{3}{x}\)

Step-by-Step Solution

Answer

(a) Not proportional; (b) inversely proportional

Step 1: Understand Direct Proportionality

Two variables, say \(y\) and \(x\), are said to be directly proportional if their ratio is constant. In mathematical terms, \(y = kx\), where \(k\) is a constant.

Step 2: Understand Inverse Proportionality

Two variables are inversely proportional if their product is constant. This means that \(y = \frac{k}{x}\), where \(k\) is a constant.

Step 3: Analyze Equation (a)

The given equation is \(y = \frac{3}{x+1}\). Here, \(y\) is expressed as \(\frac{3}{x+1}\), which does not fit the form \(kx\) (direct proportionality) or \(\frac{k}{x}\) (inverse proportionality). Therefore, \(y\) is not proportional to \(x\).

Step 4: Analyze Equation (b)

The given equation is \(y = \frac{3}{x}\). It is in the form \(\frac{k}{x}\), where \(k = 3\), which fits the definition of inverse proportionality. Thus, \(y\) is inversely proportional to \(x\).

Key Concepts

Direct Proportionality

When we talk about direct proportionality, we refer to a relationship between two variables where one variable is a constant multiple of the other. This concept is expressed mathematically as \( y = kx \). Here, \( y \) is the dependent variable, \( x \) is the independent variable, and \( k \) is the constant of proportionality. It signifies that as one variable increases, the other increases at a consistent rate. Or if one decreases, the other also decreases.

If you're given a situation or equation and asked to identify direct proportionality, look for this pattern: both variables on one side and an equal sign with just a constant in between. For example, if doubling \( x \) results in doubling \( y \), their relationship is directly proportional. This is important in many real-life applications, like speed and distance (when time is constant), or the price and quantity based on a set unit price.

Recognizing direct proportionality can simplify problem solving, as it provides flexibility for predicting how changes in one variable affect the other.

Inverse Proportionality

Inverse proportionality is another type of relationship between two variables. In this case, as one variable increases, the other decreases such that their product remains constant. This can be mathematically represented by the equation \( y = \frac{k}{x} \), where \( k \) is the constant of proportionality.

This type of relationship means that if you multiply the value of one variable by a certain factor, the other variable is divided by that same factor to maintain the constant product. For example, if \( x \) doubles, then \( y \) must be halved, provided \( k \) remains unchanged. A practical example can be found in physics, where the pressure and volume of a gas are inversely proportional at constant temperature (Boyle's Law).

When given a task to identify inverse proportionality, check if rearranging the equation would fit the form \( y = \frac{k}{x} \). This will help you recognize such relationships in mathematical problems and real-world scenarios alike.

Variables

Variables are fundamental components in mathematics and science. They are symbols, often letters, that represent unknown or changeable values. When discussing relationships like direct or inverse proportionality, variables play a crucial role.

Typically, variables like \( x \) and \( y \) can represent quantities such as time, distance, speed, and more. In equations, they help to model relationships and changes between different quantities.

Understanding variables allows us to create equations that solve problems and predict behavior based on changes in one or more aspects. For instance, in the equation \( y = 3x \), \( y \) is dependent on the value of \( x \), with the constant \( 3 \) dictating the relationship's rate of change.

By manipulating these variables, we can explore various scenarios, calculate needed values, and even derive new formulas to better understand our world. Therefore, grasping the concept of variables is foundational to mastering proportionality and other mathematical concepts.