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Chapter 4: Problem 2

Kyle said that if \(r\) is directly proportional to \(s,\) then there is some non- zero constant, \(c,\) such that \(r=c s\) and that \(\\{(s, r)\\}\) is a one-to-one function. Do you agree with Kyle? Explain why or why not.

Short Answer

Expert verified
Yes, I agree with Kyle. The function \(r = cs\) is one-to-one when \(c \neq 0\).

Step by step solution

01

Understanding Direct Proportionality

When we say that a variable \(r\) is directly proportional to another variable \(s\), it means there exists a constant \(c\) such that \(r = cs\). This relationship suggests that as \(s\) increases, \(r\) increases by the same factor, and vice versa.
02

Understanding One-to-One Function

A function is one-to-one if every element of the range of the function is mapped by a unique element of its domain. In other words, \(f(a) = f(b)\) implies that \(a = b\). This prevents two different domain values mapping to the same range value.
03

Analyze if \(r = cs\) is One-to-One

Given \(r = cs\), we can determine if it's a one-to-one function. Take two points \((s_1, r_1)\) and \((s_2, r_2)\) where \(r_1 = cs_1\) and \(r_2 = cs_2\). For the function to be one-to-one, if \(r_1 = r_2\), then \(s_1\) must equal \(s_2\). Since \(r_1 = r_2\) implies \(cs_1 = cs_2\), dividing both sides by \(c\) (assuming \(c eq 0\)) gives \(s_1 = s_2\), which satisfies the one-to-one condition.
04

Conclusion

Since \(r = cs\) implies \(s_1 = s_2\) when \(r_1 = r_2\), the relationship is indeed a one-to-one function. Therefore, I agree with Kyle's statement that \(\{(s, r)\}\) is a one-to-one function when \(r\) is directly proportional to \(s\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

One-to-One Function
A one-to-one function is a type of mathematical function where each output value is paired with one unique input value. This means that no two different input values produce the same output.
To understand this better, consider a simple example: if the function maps input 1 to output 2 and input 2 to output 3, then it qualifies as one-to-one. However, if both input 1 and input 2 map to output 3, then it isn't one-to-one.
  • Mathematically, a function \(f\) is one-to-one if for any \(a\) and \(b\) in its domain, \(f(a) = f(b)\) implies \(a = b\).
  • If a function is one-to-one, then it has an inverse function because every output can trace back to a single unique input.

    • In Kyle's case, since \(r = cs\) is formulated such that every \(r\) value is uniquely determined by its corresponding \(s\) through the constant \(c\), it verifies the condition \(s_1 = s_2\) when \(r_1 = r_2\). Hence, it is indeed a one-to-one function.
Direct Variation
Direct variation describes a relationship between two variables in which one is a constant multiple of the other. This is expressed mathematically as \(r = cs\), where \(c\) is the non-zero constant of variation, or proportionality constant.
  • If two variables, \(r\) and \(s\), are in direct variation, then as \(s\) increases, \(r\) increases by the same proportional amount.
  • The graph of a direct variation relationship is a straight line that passes through the origin \((0,0)\).

    • Understanding this concept is crucial because it implies predictability in relationships; knowing the multiplicative constant \(c\) allows for predictions of \(r\) based on any given \(s\). This application is ubiquitous in science and engineering, such as when determining speed (as a direct variation of distance and time).
Constant of Proportionality
The constant of proportionality, denoted by \(c\) in the formula \(r = cs\), represents the ratio between the two variables that maintain their constant relationship.
This constant is crucial because it embodies the exact relationship scale between the two variables, supporting their direct correlation at all points along their respective lines.
  • To find \(c\), simply divide \(r\) by \(s\) in any point of their relationship where neither is zero: \(c = \frac{r}{s}\).
  • It essentially tells you how much \(r\) changes for a unit change in \(s\).

    • The constant of proportionality isn't just a number; it provides insight into the depth of direction within the relationship between \(r\) and \(s\), facilitating a deeper understanding of this linear association. When knowing \(c\), one can quickly calculate unknown values of \(r\) given any \(s\), reinforcing the one-to-one nature of the function.

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Most popular questions from this chapter

In \(11-16,\) determine if the function has an inverse. If so, list the pairs of the inverse function. If not, explain why there is no inverse function. $$ \\{(-1,3),(-1,5),(-2,7),(-3,9),(-4,11)\\} $$

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