# Functions and Their Graphs

Site: | Saylor Academy |

Course: | MA005: Calculus I |

Book: | Functions and Their Graphs |

Printed by: | Guest user |

Date: | Tuesday, November 5, 2024, 1:45 AM |

## Description

Read this section for an introduction to functions and their graphs. Work through practice problems 1-5.

## Table of contents

- Functions and Their Graphs
- What is a Function?
- Function Machines
- Functions Defined by Equations
- Functions Defined by Graphs and Tables of Values
- Creating Graphs of Functions
- A Graph from a Table of Values
- A Graph from an Equation
- Is Every Graph the Graph of a Function?
- Reading Graphs Carefully
- Practice Problem Answers

## Functions and Their Graphs

When you prepared for calculus, you learned to manipulate functions by adding, subtracting, multiplying and dividing them, as well as calculating functions of functions (composition). In calculus, we will still be dealing with functions and their
applications. We will create new functions by operating on old ones. We will derive information from the graphs of the functions and from the derived functions. We will find ways to describe the point–by–point behavior of functions
as well as their behavior "close to" some points and also over entire intervals. We will find tangent lines to graphs of functions and areas between graphs of functions. And, of course, we will see how these ideas can be used
in a variety of fields.

This section and the next one are a review of information and procedures you should already know about functions before we begin calculus.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-1.2-Lines-in-the-Plane.pdf

This work is licensed under a Creative Commons Attribution 3.0 License.

## What is a Function?

#### Definition of Function:

A function from a set to a set is a rule for assigning to each element of the set a single element of the set . A function assigns a unique (exactly one) output element in the set to each input element from the set .

The **rule **which defines a function is often given by an equation, but it could also be given in words or graphically or by a table of values. In practice, functions are given in all of these ways, and we will use
all of them in this book.

In the definition of a function, the set of all inputs is called the **domain **of the function. The set of all outputs produced from these inputs is called the **range **of
the function. Two different inputs, elements in the domain, can be assigned to the same output, an element in the range, but one input cannot lead to 2 different outputs.

Most of the time we will work with functions whose domains
and ranges are real numbers, but there are other types of functions all around us. Final grades for this course is an example of a function. For each student, the instructor will assign a final grade based on some rule for evaluating that
student's performance. The **domain **of this function consists of all students registered for the course, and the **range **consists of the letters and perhaps (withdrawn). Two students can receive
the same final grade, but only one grade will be assigned to each student.

## Function Machines

Functions are abstract structures, but sometimes it is easier to think of them in a more concrete way. One way is to imagine that a function is a special purpose computer, a machine which accepts inputs, does something to those inputs according
to the defining rule, and produces an output. The output is the value of the function for the given input value. If the defining rule for a function is "multiply the input by itself" ,
, then Fig. 1 shows the results of putting the inputs and into the machine .

**Practice 1:** If we have a function machine g whose rule is "divide 3 by the input and add 1", , what outputs do we get from the inputs and
? What happens if we put 0 into the machine ?

You expect your calculator to behave as a function: each time you press the same input sequence of keys you expect to see the same output display.
In fact, if your calculator did not produce the same output each time you would need a new calculator. (On many calculators there is a key which does not produce the same output each time you press it. Which key is that?)

## Functions Defined by Equations

If the domain consists of a collection of real numbers (perhaps all real numbers) and the range is a collection of real numbers, then the function is called a **numerical function**. The rule for a numerical function can be given in
several ways, but it is usually written as a formula. If the rule for a numerical function, , is "the output is the input number multiplied by itself", then we could write the rule as . The use
of an "" to represent the input is simply a matter of convenience and custom. We could also represent the same function by , or .

For
the function f defined by , we have that , , and . Notice that the two different inputs, 3
and –2, both lead to the output of 6. That is allowable for a function. We can also evaluate f if the input contains variables. If we replace the "" with something else in the notation
"", then we must replace the "" with the same thing everywhere in the equation:

,

, and, in general, .

For more complicated expressions, we can just proceed step–by–step:

.

**Practice 2:** For the function defined by , evaluate , and .

## Functions Defined by Graphs and Tables of Values

The **graph **of a numerical function consists of a plot of ordered pairs where is in the domain of and . A table of values of a numerical function consists of
a list of some of the ordered pairs where . Fig. shows a graph of for .

A function can be defined by a graph or by a table of values, and
these types of definitions are common in applied fields. The outcome of an experiment will depend on the input, but the experimenter may not know the "rule" for predicting the outcome. In that case, the experimenter
usually represents the experiment function as a table of measured outcome values verses input values or as a graph of the outcomes verses the inputs. The table and graph in Fig. 3 show the deflections obtained when weights are loaded
at the end of a wooden stick. The graph in Fig. 4 shows the temperature of a hot cup of tea as a function of the time as it sits in a 68^{o} F room. In these experiments, the "rule" for the function is that actual
outcome of the experiment.

Tables have the advantage of presenting the data explicitly, but it is often difficult to detect patterns simply from lists of numbers. Graphs tend to obscure some of the precision of the data, but patterns
are much easier to detect visually - we can actually see what is happening with the numbers.

## Creating Graphs of Functions

Most people understand and can interpret pictures more quickly than tables of data or equations, so if we have a function defined by a table of values or by an equation, it is often useful and necessary to create a picture of the function, a graph.

## A Graph from a Table of Values

If we have a table of values of the function, perhaps consisting of measurements obtained from an experiment, then we can simply plot the ordered pairs in the xy–plane to get a graph which consists of a collection of points.

Fig.
5 shows the lengths and weights of trout caught (and released) during several days of fishing. It also shows a line which comes "close" to the plotted points. From the graph, you could estimate that a 17 inch trout would
weigh slightly more than one pound.

## A Graph from an Equation

Creating the graph of a function given by an equation is similar to creating one from a table of values - we need to plot enough points where so we can be confident of the shape and location of the graph of the entire
function. We can find a point which satisfies ) by picking a value for and then calculating the value for by evaluating . Then we can enter the
value in a table or simply plot the point .

If you recognize the form of the equation and know something about the shape of graphs of that form, you may not have to plot many points. If you do not recognize the
form of the equation then you will have to plot more points, maybe 10 or 20 or 234: it depends on how complicated the graph appears and on how important it is to you (or your boss) to have an accurate graph. Evaluating at a lot of different values for and then plotting the points is usually not very difficult, but it can be very time–consuming. Fortunately, there are now calculators and personal computers which will
do the evaluations and plotting for you.

## Is Every Graph the Graph of a Function?

The definition of function requires that each element of the domain, each input value, be sent by the function to exactly one element of the range, to exactly one output value, so for each input x-value there will be exactly one
output y–value, . The points and cannot both be on the graph of unless . The graphic interpretation of this result is called the Vertical Line Test.

##### Vertical Line Test for a Function:

A graph is the graph of a function if and only if a vertical line drawn through any point in the domain intersects the graph at exactly one point.

Fig.
6(a) shows the graph of a function. Figs. 6(b) and 6(c) show graphs which are not the graphs of functions, and vertical lines are shown which intersect those graphs at more than one point. Non–functions are not "bad", and sometimes they
are necessary to describe complicated phenomena.

## Reading Graphs Carefully

Calculators and computers can help students, reporters, business people and scientific professionals create graphs quickly and easily, and because of this, graphs are being used more often than ever to present information and justify arguments. And this
text takes a distinctly graphical approach to the ideas and meaning of calculus. Calculators and computers can help us create graphs, but we need to be able to read them carefully. The next examples illustrate some types of information which can be
obtained by carefully reading and understanding graphs.

**Example 1: **A boat starts from St. Thomas and sails due west with the velocity shown in Fig. 7

(a) When is the boat traveling the fastest?

(b) What does a negative velocity away from St. Thomas mean?

(c) When is the boat the farthest from St. Thomas?

Solution: (a) The greatest speed is 10 mph at .

(b) It
means that the boat is heading back toward St. Thomas.

(c) The boat is farthest from St. Thomas at . For the boat's velocity is positive, and the distance from the boat to St. Thomas is increasing. For the boat's velocity is negative, and the distance from the boat to St. Thomas is decreasing.

**Practice 3: **You and a friend start out together and hike along the

same trail but walk at different speeds (Fig. 8).

(a) Who is walking faster at ?

(b) Who is ahead at ?

(c) When are you and your friend farthest apart?

(d) Who is ahead when ?

**Example 2: **In Fig. 9, which has the largest slope: the
line through the points and , the line through and , or the line through and ?

Solution: The line through C and P has the largest slope: .

**Practice 4:** In Fig. 10, the point on the curve is fixed, and the point is moving to the right along the curve toward
the point . As moves toward :

(a) the x–coordinate of is Increasing, Decreasing, Remaining constant, or None of these.

(b) the x–increment from to is Increasing, Decreasing, Remaining constant, or None of these

(c) the slope from to is Increasing, Decreasing, Remaining constant, or None of these.

**Example 3:** The graph of is shown in Fig. 11. Let be the slope of the line tangent to the graph of
at the point .

(a) Estimate the values , and .

(c) At what value(s) of is largest?

(d) Sketch the graph of .

Solution: (a) Fig. 11 shows the graph with several tangent lines to the graph of . From Fig. 11 we can estimate that (the slope of the line tangent to the graph of f at (1,0) )
is approximately equal to 1. Similarly, and .

(b) The slope of the tangent line appears to be horizontal () at and at .

(c) The tangent line to the graph appears to have greatest slope
(be steepest) near .

(d) We can build a table of values of and then sketch the graph of these values.

x | f(x) | g(x) = tangent slope at (x, f(x) ) |
---|---|---|

0 | -1 | .5 |

1 | 0 | 1 |

2 | 2 | 0 |

3 | 1 | -1 |

4 | 0 | -1 |

5 | -1 | 0 |

6 | -.5 | .5 |

**Practice 5:**Water is flowing into a container (Fig. 13) at a constant rate of 3 gallons per minute. Starting with an empty container, sketch the graph of the height of the water in the container as a function of time.

## Practice Problem Answers

**Practice 1: **

Input | Output |
---|---|

5 | |

0 | which is not defined because of division by 0. |

Input | Output |
---|---|

**Practice 2:** .

. .

.

.

.

.

**Practice 3:** (a) Friend (b) Friend (c) At . Before that your friend is walking faster and increasing the distance
between you. Then you start to walk faster than your friend and start to catch up. (d) Friend. You are walking faster than your friend at , but you still have not caught up.

**Practice 4: **(a)
The x–coordinate is increasing. (b) The x–increment is decreasing. (c) The slope of the line through and is decreasing.

**Practice 5: **See
Fig. 31.