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In this section, you will:
  • Determine whether a relation represents a function.
  • Find the value of a function.
  • Determine whether a function is one-to-one.
  • Use the vertical line test to identify functions.
  • Graph the functions listed in the library of functions.

A jetliner changes altitude as its distance from the starting point of a flight increases. The weight of a growing child increases with time. In each case, one quantity depends on another. There is a relationship between the two quantities that we can describe, analyze, and use to make predictions. In this section, we will analyze such relationships.

Determining whether a relation represents a function

A relation    is a set of ordered pairs. The set consisting of the first components of each ordered pair    is called the domain and the set consisting of the second components of each ordered pair is called the range . Consider the following set of ordered pairs. The first numbers in each pair are the first five natural numbers. The second number in each pair is twice that of the first.

{ ( 1 , 2 ) , ( 2 , 4 ) , ( 3 , 6 ) , ( 4 , 8 ) , ( 5 , 10 ) }

The domain is { 1 , 2 , 3 , 4 , 5 } . The range is { 2 , 4 , 6 , 8 , 10 } .

Note that each value in the domain is also known as an input value, or independent variable    , and is often labeled with the lowercase letter x . Each value in the range is also known as an output value, or dependent variable    , and is often labeled lowercase letter y .

A function f is a relation that assigns a single element in the range to each element in the domain . In other words, no x -values are repeated. For our example that relates the first five natural numbers    to numbers double their values, this relation is a function because each element in the domain, { 1 , 2 , 3 , 4 , 5 } , is paired with exactly one element in the range, { 2 , 4 , 6 , 8 , 10 } .

Now let’s consider the set of ordered pairs that relates the terms “even” and “odd” to the first five natural numbers. It would appear as

{ ( odd , 1 ) , ( even , 2 ) , ( odd , 3 ) , ( even , 4 ) , ( odd , 5 ) }

Notice that each element in the domain, { even, odd } is not paired with exactly one element in the range, { 1 , 2 , 3 , 4 , 5 } . For example, the term “odd” corresponds to three values from the domain, { 1 , 3 , 5 } and the term “even” corresponds to two values from the range, { 2 , 4 } . This violates the definition of a function, so this relation is not a function.

[link] compares relations that are functions and not functions.

Three relations that demonstrate what constitute a function.
(a) This relationship is a function because each input is associated with a single output. Note that input q and r both give output n . (b) This relationship is also a function. In this case, each input is associated with a single output. (c) This relationship is not a function because input q is associated with two different outputs.

Function

A function    is a relation in which each possible input value leads to exactly one output value. We say “the output is a function of the input.”

The input    values make up the domain    , and the output    values make up the range    .

Given a relationship between two quantities, determine whether the relationship is a function.

  1. Identify the input values.
  2. Identify the output values.
  3. If each input value leads to only one output value, classify the relationship as a function. If any input value leads to two or more outputs, do not classify the relationship as a function.

Questions & Answers

sin^4+sin^2=1, prove that tan^2-tan^4+1=0
SAYANTANI Reply
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"
Kuz Reply
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Kuz
A = P(1 + r/n) ^rt
Dale
how to solve an expression when equal to zero
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its a very simple
Kavita
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Tomorrow its an revision on factorising and Simplifying...
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Masum
What is the value of log-1
Masum
the value of log1=0
Kavita
Log(-1)
Masum
What is the value of i^i
Masum
log -1 is 1.36
kurash
No
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Kavita
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Kavita
tan20°×tan30°×tan45°×tan50°×tan60°×tan70°
Joju Reply
jaldi batao
Joju
Find the value of x between 0degree and 360 degree which satisfy the equation 3sinx =tanx
musah Reply
what is sine?
tae Reply
what is the standard form of 1
Sanjana Reply
1×10^0
Akugry
Evalute exponential functions
Sujata Reply
30
Shani
The sides of a triangle are three consecutive natural number numbers and it's largest angle is twice the smallest one. determine the sides of a triangle
Jaya Reply
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Inkoom
3, 4, 5 principle from geo? sounds like a 90 and 2 45's to me that my answer
Neese
answer is 2, 3, 4
Gaurav
prove that [a+b, b+c, c+a]= 2[a b c]
Ashutosh Reply
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Akugry
i can prove [a+b+b+c+c+a]=2[a+b+c]
this is simple
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Stormzy
x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
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Oliver
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Kwesi Reply
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Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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