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1.1 Functions and function notation

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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 of the first components of each ordered pair is called the domain and the set 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 value in the range to each value 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

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
ADNAN Reply
linear speed of an object
Melissa Reply
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
Melissa
test
Matrix
how to find domain
Mohamed Reply
like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
Dan
define the term of domain
Moha
if a>0 then the graph is concave
Angel Reply
if a<0 then the graph is concave blank
Angel
what's a domain
Kamogelo Reply
The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.
Spiro
how fast can i understand functions without much difficulty
Joe Reply
what is inequalities
Nathaniel
functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2
Dan
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49
Dan
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2
Dan
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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