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Chapter 3: Problem 61

Consider the following two rules, \(F\) and \(G,\) where \(F\) is the rule that assigns to each person his or her birth-mother and \(G\) is the rule that assigns to each person his or her aunt. Explain why \(F\) is a function but \(G\) is not.

Short Answer

Expert verified
Rule F is a function while Rule G is not because Rule F maps each person to one birth-mother, but Rule G can map a person to multiple aunts.

Step by step solution

01

Define a Function

A function is a relation from a set of inputs to a set of possible outputs where each input is related to exactly one output. In simpler terms, each element in the domain must map to one and only one element in the co-domain.
02

Analyze Rule F

Rule F assigns to each person their birth-mother. Since a person can have only one birth-mother, each person (input) is mapped to exactly one birth-mother (output). Therefore, Rule F satisfies the definition of a function.
03

Analyze Rule G

Rule G assigns to each person their aunt. A person can have multiple aunts, for example, through different sisters of their parents. Therefore, a single person (input) can be associated with more than one aunt (output), violating the requirement for exactly one output for each input.
04

Conclusion

Since Rule F assigns each person to one specific birth-mother, it fulfills the requirement of a function. However, Rule G can assign multiple outputs (aunts) to a single input (person), which disqualifies it as a function.

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Key Concepts

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

Definition of Function
In mathematics, a function is a set of rules that uniquely assigns an output to each input from a designated set of inputs, known as its domain.
For a relation to qualify as a function, each input must be paired with exactly one output.
This ensures that there is a clear and predictable relationship between the input and the output.
  • The set of all possible inputs is called the domain.
  • The set of all possible outputs is referred to as the co-domain.
  • When each input is mapped to one output, we call this relationship a function.
It's helpful to think of functions as machines that take inputs, process them according to a specific rule, and then provide a single output for each input.
Input-Output Relation
The input-output relationship is at the core of understanding functions. This determines how each element in the input set (domain) associates with an element in the output set (co-domain).
The essential rule is that any single input can produce no more than one output, cementing the idea of predictability and consistency.
  • Consider rule F, which assigns every individual to their birth-mother. Since every person has only one biological birth-mother, each input (person) has a precise output (birth-mother).
  • However, rule G, assigning individuals to their aunts, lacks uniqueness. A person might have multiple aunts, showing inconsistency and making the rule fail as a function.
This illustrates why exact one-to-one pairing between inputs and outputs is necessary for a function.
Mapping
Mapping is a fundamental term in describing how functions work, focusing on the journeys of inputs to their designated outputs within a function's framework.
It means linking each input in the domain to one specific output in the co-domain. This connection is integral to defining a function.
  • Rule F's mapping to a single birth-mother for each person epitomizes ideal function mapping, as each person's input finds one exclusive spot in the output domain.
  • In contrast, rule G's multiple possible links with aunts exemplify a breach in functional mapping, with one input branching into several outputs, disrupting function viability.
Mapping therefore reinforces the exact one-to-one relationship principle, underscoring the functional structure of well-defined mathematical rules.

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Most popular questions from this chapter

Let \(f(x)=\frac{x-3}{x+c},\) where \(c\) denotes a constant. (a) If \(c=-1,\) show that \(f(f(x))=x\) (b) Use a graphing utility to support the result in part (a). That is, enter the function \(f(x)=\frac{x-3}{x-1},\) and then have the machine graph the two functions \(y=x\) and \(y=f(f(x))\) in the same picture. (c) What does the result in part (a) tell you about the iteration process for the function \(y=(x-3) /(x-1) ?\) That is, what pattern emerges in the iterates? Answer in complete sentences. (d) Now assume \(c=1,\) instead of \(-1 .\) Show that \(f(f(f(x)))=x\). (e) Use a graphing utility to support the result in part (d). (f) What does the result in part (d) tell you about the iteration process for the function \(y=(x-3) /(x+1) ?\) That is, what pattern emerges in the iterates? Answer in complete sentences.

Suppose that the demand function for a certain item is given by \(p(x)=\frac{24}{2^{x / 100}},\) where \(x\) is the number of items that can be sold when the price of each item is \(p(x)\) dollars. Compute \(\Delta p / \Delta x\) over each of the intervals \(0 \leq x \leq 100\) and \(300 \leq x \leq 400 .\) Include units in your answers. Why does it make sense (economically) that the answers are negative?

Suppose that \((a, b)\) is a point on the graph of \(y=f(x)\) Match the functions defined in the left-hand column with the points in the right-hand column. For example, the appropriate match for (a) in the left-hand column is determined as follows. The graph of \(y=f(x)+1\) is obtained by translating the graph of \(y=f(x)\) up one unit. Thus the point \((a, b)\) moves up to \((a, b+1)\), and consequently, (E) is the appropriate match for (a). (a) \(y=f(x)+1 \quad\) (A) \((-a, b)\) (b) \(y=f(x+1)\) (B) \((b, a)\) (c) \(y=f(x-1)+1\) (C) \((a-1, b)\) (d) \(y=f(-x)\) (D) \((-b, a+1)\) (e) \(y=-f(x)\) (E) \((a, b+1)\) (f) \(y=-f(-x)\) (F) \((1-a, b)\) (g) \(y=f^{-1}(x)\) (G) \((-a,-b)\) (h) \(y=f^{-1}(x)+1\) (H) \((-b, 1-a)\) (i) \(y=f^{-1}(x-1)\) (I) \((b,-a)\) (j) \(y=f^{-1}(-x)+1\) (J) \((a,-b)\) (k) \(y=-f^{-1}(x)\) (K) \((b+1, a)\) (1) \(\quad y=-f^{-1}(-x)+1\) (L) \((a+1, b+1)\) (m) \(y=1-f^{-1}(x)\) (M) \((b, a+1)\) (n) \(y=f(1-x)\) (N) \((b, 1-a)\)

(a) Complete the given tables. (Use a calculator where necessary.) $$\begin{array}{lllllll} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\\\\sqrt{x} & & & & & \\\\\hline & & & & & \\\\\hline x & 0 & -1 & -2 & -3 & -4 & -5 \\\\\ \sqrt{-x} & & & & & \\\\\hline\end{array}$$ (b) Using the tables, graph the functions \(y=\sqrt{x}\) and \(y=\sqrt{-x}\) on the same set of axes. How are the graphs related?

Let \(q(x)=a x^{2}+b x+c .\) Evaluate $$q\left(\frac{-b+\sqrt{b^{2}-4 a c}}{2 a}\right)$$
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