Problem 7 Select all of the following tabl... [FREE SOLUTION]

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Precalculus: An Investigation of Functions

David Lippman

$Math Studyset 91影视 Explanations$ Math

2017 Edition

Chapter 1: Problem 7

Select all of the following tables which represent $y$ as a function of $x$. a. $$ \begin{array}{|l|l|l|l|} \hline \boldsymbol{x} & 5 & 10 & 15 \\ \hline \boldsymbol{y} & 3 & 8 & 14 \\ \hline \end{array} $$ b. $$ \begin{array}{|l|l|l|l|} \hline \boldsymbol{x} & 5 & 10 & 15 \\ \hline \boldsymbol{y} & 3 & 8 & 8 \\ \hline \end{array} $$ c. $$ \begin{array}{|l|l|l|l|} \hline x & 5 & 10 & 10 \\ \hline y & 3 & 8 & 14 \\ \hline \end{array} $$

Short Answer

Expert verified

Tables a and b represent functions, table c does not.

Step by step solution

Understanding a Function

A table represents a function if every input value of $x$ corresponds to exactly one output value of $y$. This means for each $x$ there is only one $y$ value.

Analyze Table a

In table a, $x = 5$ maps to $y = 3$, $x = 10$ maps to $y = 8$, and $x = 15$ maps to $y = 14$. Each $x$ value has a unique $y$ value. Table a represents a function.

Analyze Table b

In table b, $x = 5$ maps to $y = 3$, $x = 10$ maps to $y = 8$, and $x = 15$ also maps to $y = 8$. Each $x$ value has a unique $y$ value. Table b represents a function.

Analyze Table c

In table c, $x = 5$ maps to $y = 3$, but $x = 10$ maps to both $y = 8$ and $y = 14$. There are two different $y$ values for the same $x = 10$, so it does not represent a function.

Conclusion

Tables a and b represent $y$ as a function of $x$ because each $x$ has only one $y$ value, whereas table c does not meet this criterion.

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Key Concepts

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

Function definition

A function is a special kind of relationship between two sets, typically referred to as inputs and outputs. In mathematical terms, a function is defined by a rule that assigns each input exactly one output. The input values are often denoted as "x" and the output values as "y." To identify a function, each input should map to only a single output. If every input does indeed produce one unique output, then what you have is a function.
Understanding this allows us to analyze whether given data represents a function or not.

Mapping

In the context of functions, mapping refers to how each input in one set is related to an output in another set. Let's say we have a set of all possible "x" values and another set of "y" values. When we talk about mapping, we're looking at how each "x" is paired with a "y."
A helpful way to visualize this is by thinking of each "x" value having a single arrow pointing to exactly one "y" value.

If one arrow points to multiple "y" values, it鈥檚 not a function.
Conversely, if each "x" has only one arrow pointing to a "y," then it is a valid mapping that defines a function.

This ensures that a function is a consistent and predictable operation.

Unique output for input

A fundamental property of functions is the idea of a unique output for each input. This notion is what distinctly defines a function from other types of relations. If an input "x" can correspond to multiple outputs "y," it fails to be a function.
Think of this in terms of a machine: for every input item inserted, it should produce only one result.
For instance, in the context of a function presented in a table, each "x" value should have only one corresponding "y" value listed beside it.

Table a from the problem shows this rule well as each "x" has a unique "y."
If you ever encounter an input mapped to two different outputs, like in table c, it's a clear sign that the relation is not a function.

This rule ensures clarity and predictability when working with functions.

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91影视

Short Answer

Step by step solution

Understanding a Function

Analyze Table a

Analyze Table b

Analyze Table c

Conclusion

Key Concepts

Function definition

Mapping

Unique output for input

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