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Chapter 3: Problem 7

Determine whether each relation is also a function. $$ \\{(-1,0),(-1,6),(-1,8)\\} $$

Short Answer

Expert verified
This relation is not a function because the input -1 corresponds to multiple outputs (0, 6, and 8).

Step by step solution

01

Understand the Relation

A relation is a set of ordered pairs. In this exercise, the relation is \((-1,0), (-1,6), (-1,8)\). Each pair consists of a unique input (first element) and output (second element).
02

Define a Function

A function is a special type of relation where each input has exactly one unique output. If any input corresponds to more than one output, the relation is not a function.
03

Analyze the Inputs

Check all the inputs in the relation. In this case, the input is \(-1\) for all pairs. Therefore, the inputs are not unique.
04

Check for Multiple Outputs

Since all inputs are \(-1\), we look at their outputs: 0, 6, and 8. An input having multiple outputs indicates that this is not a function.

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Key Concepts

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

Relations and Functions
In algebra, understanding the concept of relations and functions is essential as they form the backbone of many mathematical operations. A relation is a collection of ordered pairs where each element in the first set is associated with elements in the second set. Think of it as a link between two things.
In contrast, a function is a refined form of relation where each input has exactly one output. In simpler terms, if you input a value into a function, you will always get the same result out. This unique characteristic differentiates functions from general relations. Ensuring that each input contributes to a single output is crucial in defining it as a function.
Ordered Pairs
Ordered pairs are fundamental in mathematics to represent points or elements in any contextual relation. Each ordered pair is structured as \(x, y\), where \(x\) is the input, and \(y\) is the corresponding output. Clarity on this format allows a simplification of interpreting complex data.
Through ordered pairs, we can easily map and analyze relationships between two data sets. For instance, in the set \{(-1,0), (-1,6), (-1,8)\}, each bracket represents an ordered pair that connects a specific input with an output. This notation is instrumental when identifying how inputs relate to outputs and is practical for graph plotting.
Unique Input
A unique input is a key criterion in defining a function. Each input value of a function must stand out with only one corresponding output. It ensures consistency and predictability in mathematical operations.
For example, imagine entering a unique key on a piano: it should always produce the same note. Similarly, in mathematics, an input should not serve different outputs. If all inputs within a relation are unique, each can comfortably map to one specific output, classifying it as a function.
Multiple Outputs
Multiple outputs signify that one input gives rise to more than one result, which violates the definition of a function. This scenario occurs when a single input correlates with varying outputs.
For instance, the relation \{(-1,0), (-1,6), (-1,8)\} shows that input \(-1\) links to three different outputs: 0, 6, and 8. This multiplicity ceases the ability to classify the relation as a function since a necessary condition is having only one output per input.
  • If you discover multiple outputs for an individual input, it signals that the relation fails the function test. This distinction is crucial in identifying and characterizing mathematical relationships.

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Most popular questions from this chapter

See Examples 1 through 7 . Find an equation of each line described. Write each equation in slope-intercept form (solved for \(y\) ), when possible. With slope 0, through \((6.7,12.1)\)

Write an ordered pair for each point described. Point \(D\) is three units to the left of the origin.

Solve. Assume each exercise describes a linear relationship. Write the equations in slope-intercept form. See Example 8 . The birth rate in the United States in 1996 was 14.7 births per thousand population. In 2006 , the birth rate was 14.14 births per thousand. (Source: Department of Health and Human Services, National Center for Health Statistics) CAN'T COPY THE IMAGE a. Write two ordered pairs of the form (years after 1996 birth rate per thousand population). b. Assume that the relationship between years after 1996 and birth rate per thousand is linear over this period. Use the ordered pairs from part (a) to write an equation of the line relating years to birth rate. c. Use the linear equation from part (b) to estimate the birth rate in the United States in the year 2016 .

Write an equation in standard form of the line that contains the point \((-1,2)\) and is a. parallel to the line \(y=3 x-1\) b. perpendicular to the line \(y=3 x-1\)

Example $$\text { If } f(x)=x^{2}+2 x+1, \text { find } f(\pi)$$ Solution: $$\begin{aligned} &f(x)=x^{2}+2 x+1\\\ &f(\pi)=\pi^{2}+2 \pi+1 \end{aligned}$$ Given the following functions, find the indicated values. $$f(x)=x^{2}-12$$ a. \(f(12)\) b. \(f(a)\)
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