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Find the domain and range for f ( x ) = 4 2 x + 5 .

Domain = { x | x 2 } , range = { y | y 5 }

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Representing functions

Typically, a function is represented using one or more of the following tools:

  • A table
  • A graph
  • A formula

We can identify a function in each form, but we can also use them together. For instance, we can plot on a graph the values from a table or create a table from a formula.

Tables

Functions described using a table of values    arise frequently in real-world applications. Consider the following simple example. We can describe temperature on a given day as a function of time of day. Suppose we record the temperature every hour for a 24-hour period starting at midnight. We let our input variable x be the time after midnight, measured in hours, and the output variable y be the temperature x hours after midnight, measured in degrees Fahrenheit. We record our data in [link] .

Temperature as a function of time of day
Hours after Midnight Temperature ( ° F ) Hours after Midnight Temperature ( ° F )
0 58 12 84
1 54 13 85
2 53 14 85
3 52 15 83
4 52 16 82
5 55 17 80
6 60 18 77
7 64 19 74
8 72 20 69
9 75 21 65
10 78 22 60
11 80 23 58

We can see from the table that temperature is a function of time, and the temperature decreases, then increases, and then decreases again. However, we cannot get a clear picture of the behavior of the function without graphing it.

Graphs

Given a function f described by a table, we can provide a visual picture of the function in the form of a graph. Graphing the temperatures listed in [link] can give us a better idea of their fluctuation throughout the day. [link] shows the plot of the temperature function.

An image of a graph. The y axis runs from 0 to 90 and has the label “Temperature in Fahrenheit”. The x axis runs from 0 to 24 and has the label “hours after midnight”. There are 24 points on the graph, one at each increment of 1 on the x-axis. The first point is at (0, 58) and the function decreases until x = 4, where the point is (4, 52) and is the minimum value of the function. After x=4, the function increases until x = 13, where the point is (13, 85) and is the maximum of the function along with the point (14, 85). After x = 14, the function decreases until the last point on the graph, which is (23, 58).
The graph of the data from [link] shows temperature as a function of time.

From the points plotted on the graph in [link] , we can visualize the general shape of the graph. It is often useful to connect the dots in the graph, which represent the data from the table. In this example, although we cannot make any definitive conclusion regarding what the temperature was at any time for which the temperature was not recorded, given the number of data points collected and the pattern in these points, it is reasonable to suspect that the temperatures at other times followed a similar pattern, as we can see in [link] .

An image of a graph. The y axis runs from 0 to 90 and has the label “Temperature in Fahrenheit”. The x axis runs from 0 to 24 and has the label “hours after midnight”. There are 24 points on the graph, one at each increment of 1 on the x-axis. The first point is at (0, 58) and the function decreases until x = 4, where the point is (4, 52) and is the minimum value of the function. After x=4, the function increases until x = 13, where the point is (13, 85) and is the maximum of the function along with the point (14, 85). After x = 14, the function decreases until the last point on the graph, which is (23, 58). A line connects all the points on the graph.
Connecting the dots in [link] shows the general pattern of the data.

Algebraic formulas

Sometimes we are not given the values of a function in table form, rather we are given the values in an explicit formula. Formulas arise in many applications. For example, the area of a circle of radius r is given by the formula A ( r ) = π r 2 . When an object is thrown upward from the ground with an initial velocity v 0 ft/s, its height above the ground from the time it is thrown until it hits the ground is given by the formula s ( t ) = −16 t 2 + v 0 t . When P dollars are invested in an account at an annual interest rate r compounded continuously, the amount of money after t years is given by the formula A ( t ) = P e r t . Algebraic formulas are important tools to calculate function values. Often we also represent these functions visually in graph form.

Given an algebraic formula for a function f , the graph of f is the set of points ( x , f ( x ) ) , where x is in the domain of f and f ( x ) is in the range. To graph a function given by a formula, it is helpful to begin by using the formula to create a table of inputs and outputs. If the domain of f consists of an infinite number of values, we cannot list all of them, but because listing some of the inputs and outputs can be very useful, it is often a good way to begin.

Questions & Answers

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aparna Reply
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funny
I don't understand the formula
Adaeze Reply
who's formula
funny
What is a independent variable
Sifiso Reply
a variable that does not depend on another.
Andrew
solve number one step by step
bil Reply
x-xcosx/sinsq.3x
Hasnain
x-xcosx/sin^23x
Hasnain
how to prove 1-sinx/cos x= cos x/-1+sin x?
Rochel Reply
1-sin x/cos x= cos x/-1+sin x
Rochel
how to prove 1-sun x/cos x= cos x / -1+sin x?
Rochel
how to prove tan^2 x=csc^2 x tan^2 x-1?
Rochel Reply
divide by tan^2 x giving 1=csc^2 x -1/tan^2 x, rewrite as: 1=1/sin^2 x -cos^2 x/sin^2 x, multiply by sin^2 x giving: sin^2 x=1-cos^2x. rewrite as the familiar sin^2 x + cos^2x=1 QED
Barnabas
how to prove sin x - sin x cos^2 x=sin^3x?
Rochel Reply
sin x - sin x cos^2 x sin x (1-cos^2 x) note the identity:sin^2 x + cos^2 x = 1 thus, sin^2 x = 1 - cos^2 x now substitute this into the above: sin x (sin^2 x), now multiply, yielding: sin^3 x Q.E.D.
Andrew
take sin x common. you are left with 1-cos^2x which is sin^2x. multiply back sinx and you get sin^3x.
navin
Left side=sinx-sinx cos^2x =sinx-sinx(1+sin^2x) =sinx-sinx+sin^3x =sin^3x thats proved.
Alif
how to prove tan^2 x/tan^2 x+1= sin^2 x
Rochel
not a bad question
Salim
what is function.
Nawaz Reply
what is polynomial
Nawaz
an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Alif
a term/algebraic expression raised to a non-negative integer power and a multiple of co-efficient,,,,,, T^n where n is a non-negative,,,,, 4x^2
joe
An expression in which power of all the variables are whole number . such as 2x+3 5 is also a polynomial of degree 0 and can be written as 5x^0
Nawaz
what is hyperbolic function
vector Reply
find volume of solid about y axis and y=x^3, x=0,y=1
amisha Reply
3 pi/5
vector
what is the power rule
Vanessa Reply
Is a rule used to find a derivative. For example the derivative of y(x)= a(x)^n is y'(x)= a*n*x^n-1.
Timothy
how do i deal with infinity in limits?
Itumeleng Reply
Add the functions f(x)=7x-x g(x)=5-x
Julius Reply
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
Cabdalla Reply
difference btwn domain co- domain and range
Cabdalla
x
Verna
The set of inputs of a function. x goes in the function, y comes out.
Verna
where u from verna
Arfan
If you differentiate then answer is not x
Raymond
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
Champro
what is functions
mahin Reply
give different types of functions.
Paul

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Source:  OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2
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