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Depth of Content Knowledge: Functions


Sarah Henderson

Appalachian State University


Abstract

This paper explores the history, characteristics and the use of functions. The paper dives into

detail about the specifics of linear, exponential and rational functions. There are many different

resources used for this paper. There are scholarly articles, books on the history of mathematics,

and math education online resources. This paper is written for a second level Introduction to

Structure of Mathematics class to go in depth of our knowledge about functions.


The introduction of functions was fundamental movement for mathematics because

became widely used in all branches of math, but also spread over many different areas in science,

engineering, communications and businesses stimulating further research and knowledge in all

areas. Functions are composed of many different mathematical ideas of relationships between

two quantities that range from counting to using square roots and reciprocals. Around the early

18th century, the concept of a function in mathematics stemmed from the idea of polynomials

(sums of quantities such as 2x, 4 x2 , etc., each of which is an algebraic quantity raised to some

power and multiplied by some number (coefficient)) because “the numerical value of the sum

depends on the value of x so that the sum varies as x varies.” (Motz, Weaver, 1993, pg. 63) In

the 1730’s, a Swiss mathematician, Leonhard Euler used his prior knowledge of differential

equations to form a definition of a function describing it as “a variable quantity is an analytic

expression composed in any way whatsoever of the variable quantity and numbers or constant

quantities” (Katz, 2009, pg. 618). From this knowledge and the use of proven algebraic concepts

in the late 17th century, many different mathematicians spent a lot of their lives researching and

studying the concept of a function. The definition of a function constantly changed over the next

centuries with the incorporation of new information and ideas. Gottfriend Wilhelm Leibnitz was

the first mathematician to coin the term ‘function’ in 1673 where it was described as “a term to

represent quantities that were dependent on one variable by means of analytical expression”

(Ponte, pg. 2). After receiving help from Johann Bernoulli, a Swiss mathematician, the term

‘function’ was published with the definition “a variable as a quantity that is composed in some

way from that variable and constants.” (Ponte). Not until the 19th century did the definition of a

function undergo changes to make it more efficient and clear. During this time, mathematicians

wanted it to be known that a function should be regarded as a solution of a problem. In 1837


another mathematician, Peter Gustav Lejeune Dirichlet used the work from his previous

successors and new knowledge to have more emphasis on the correspondence between variables

representing numerical sets. From this, “a function, then, became a correspondence between two

variables so that to any value of the independent variable, there is associated one and only one

value of the dependent variable.” After this definition was proposed, mathematicians felt the

need to include all arbitrary correspondences to accommodate for unique sets with non-numbers.

From this point on, the definition of a function continued to change and become more accurate to

what mathematicians felt was necessary. Adolph P. Yushkevitch, a mathematician who

published over 300 works in mathematics stated “It was the analytical method of introducing

functions that revolutionized mathematics and, because of its extraordinary efficiency, secured a

central place for the notion of function in all exact sciences.” (Foerster, 2005, pg. 13)

A pre-calculus textbook stated that “A function f from a set A to a set B is a relation that

assigns to each element x in the set A exactly one element y in the set B. The set A is the domain

(or set of inputs) of the function f, and the set B contains the range (or set of outputs)” (Larson,

Hostetler, 2007, pg. 35). The book goes on to elaborate the different characteristics of a function

from Set A to Set B discussing that each element in A must be matched with an element in B,

some elements in B may not be matched with any element in A, two or more elements in A may

be matched with the same element, and finally an element in A (the domain) cannot be matched

with two different elements in B. (Larson, Hostetler, 2007, p. 40). In summary, there can only

be one element of the range for an individual element of the domain, but two domains may have

the same range. This specific textbook discusses how there are four individual ways that you can

represent a function: verbally, numerically, graphically, or algebraically. You can represent a

function verbally by explaining how the domain is related to the range, numerically by placing


the input and output values in a table to show their relationship, graphically by using the function

notation, y= f(x), where y equals the value of f at x, and algebraically by using the function

notation with the two variables. By laying out all the elements that make up a function it will

help students understand all the aspects. This is crucial because functions can be used for so

many different situations in mathematics. They can be used to solve real world problems, use a

pattern to determine what number will come next in a sequence, or describe a relationship

between two groups of numbers. This textbook covered the basics of functions that would help

students get an understanding of its importance.

Looking up functions in the index of a calculus textbook, there are thirty-six different sub

categories. The section where the textbook explains the basic understanding of function is they

go straight into describing what each part of the function notation represents and how that is used

in linear, quadratic, polynomial, power, exponential, rational algebraic, absolute value, and

trigonometric or circular functions and skip over the exact definition of what a function is using

domain and range. The objective of the beginning of the function chapter is described as “Given

a function y= f(x) specified by a graph, a table of values, or an equation, describe whether the y-

value is increasing or decreasing as x increases through a particular value, and estimate the

instantaneous rate of change of y at that value of x.” (Foerster, 2010, pg. 6). Going into calculus

they are expecting you to understand what a function is, and the different elements that it is

composed of. Very soon after the objective of the section, they go into describing a derivative

and limits, which are both using functions in a much different and higher thinking way than

described in the pre-calculus book. Without knowing the information in the pre-calculus book,

we would be unable to perform the functions described in the calculus book. Throughout

mathematics in school, functions continue to build on each other adding more and more complex


ideas to get different answers. From one function you are able to obtain a lot of information

depending on the way you choose to evaluate.

Linear functions can be described by an arithmetic sequence. In class, we call a sequence

arithmetic “when the common difference between successive terms of a sequence is always the

same number.” One characteristic of linear functions is that the ratio between the change in

output and change in input is always constant. So the rate of change between successive terms

will always be the same. A linear function will always be represented by a straight line due to

the constant difference. The line can be negative (slope < 0) or positive (slope > 0). When we

are looking at two groups of data, the easiest way to tell if it is a linear model is to find the

differences between the successive terms. If the constant change in inputs corresponds to a

constant change in output, then we know this a linear function. For example:

For every increase of one represented in the X value column, the Y value increases by 8.

This is true for every successive term to follow, and you could prove this by finding the

difference between the terms to make sure they are constant. For a arithmetic sequence, you

have to make sure that the X values are increasing by the same constant term as

well or else the data for your Y values may be skewed.

Although there is a pattern represented in this table, the increase in the Y axis is

different. From 2 to 4 there is a difference of 2, from 4 to 9 there is a difference of

5, from 9 to 16 there is a difference of 7. This is not a common difference

therefore it could not be identified as having an arithmetic sequence.

X Y

0 0

1 8

2 16

3 24

4 32

5 40

6 48X Y

1 2

2 4

3 9

4 16

5 25

6 36

7 49


If we were to graph the first set of data, we would get a straight line. One of the main

characteristics of a linear function can be represented by a straight line on a graph. A linear

function takes the form f(x) = mx +b, where the slope is represented as m, and the y- intercept is

represented as b. We can find our slope from using our common difference between the terms in

the arithmetic sequence. By using this formula, and knowing the slope, we can find individual

points on the graph, because the rate of change is constant, so the terms will always be increasing

or decreasing the same amount. For the linear chart above our linear equation would be: y= 8x.

Our slope is 8, and then we find our Y intercept where X is 0, and for this set of data, our Y is 0

as well. Most word problems use the word ‘constant’ in their statement. For example: The car

was driving at a constant speed on the highway for two hours, if he was going 60 miles per hour,

how long will it take him to drive 270 miles? So since we know that he will always be going 60

miles per hour, we will be able to calculate the needed information based off a linear model. For

the first hour, he would drive 60 miles, the second hour he would drive 60 more miles, so we

would be at 120 miles, then another hour he would be at 180 miles, the fourth hour, 240 miles,

and by the fifth hour he would have driven 300 miles. So to drive 270 miles it would take him

about 4 and a half hours, since 270 is right in the middle of 240 and 300. We could also insert

this information into our equation. So we know our slope (or constant change) is 60, and he is

starting at 0 miles, so our equation would be y=60x. To find how many hours it takes, we would

plug 270 in for y, since that is the amount of miles he will be traveling. So our new equation is

270= 60x, and from here we solve for x. So we would have 270/60=x. and we get that x= 4.5

hours.


An exponential function can be described by a geometric sequence. As described in

class, a geometric sequence is when the common ratio between successive terms of a sequence is

always constant, when every input (X) has a single output (Y). In order to have a common ratio,

we need to use multiplication or division, because we cannot find a ratio using addition or

subtraction. An exponential function is represented in a graph as starting out growing or

decaying slowly and then it begins to increase or decrease extremely quickly because it is

increasing or decreasing by an exponential number. A set of terms that satisfy the needs of an

exponential function could be:

For this scenario, we have to find a ratio that is consistent between the sequences of numbers. So

since the X values are all constant (increasing by 1), we are able to find

the ratio between the Y values. As we can see there is not a common

difference between the successive terms therefore it could be an

exponential function, if we have a common ratio. This one it is pretty

simple to see that we are going to divide by 2 every time to get the next

term. Our ratio between consecutive terms is ½. By checking to see if

there is a common difference first, it makes it easier to see if the function

is going to be exponential or linear.

When we use the terms in a geometric sequence we can start to set up an exponential

function. An exponential function is represented in the equation form of f ( x )=abx. Where a is

the initial value and has to be a real number, b is the growth factor which is a positive real

number and x is your exponent and/ or the ratio. To put this formula into practice, we will look at

the chart uses above. We noted that we were dividing by 2 to get to each successive term, so to

put that in ratio terms, we would be multiplying each term by ½. So this would be our growth

X Y Diff

0 64 -32

1 32 -16

2 16 -8

3 8 -4

4 4 -2

5 2 -1

6 1


factor in the equation. Our initial value is the Y value when X is 0. So for this chart our initial

value would be 64. Our final equation for this set of numbers is g ( x )=64 × 1/2x.

Domain is defined as “the set of all possible input values (often the “x” variable), which

produce a valid output from a particular function. The domain is represented by a set of ‘input’

values. The domain for a linear function is the set of all real numbers for which a function is

mathematically defined. Most often for a linear function, the domain is all real numbers, since

any real number can be substituted for x. In exponential functions, the domain could also be all

real numbers. The domain for any particular graph would be all of the x-values that are

represented in the graph.

Range is defined as “the set of all possible output values (usually the variable y, or

sometimes expressed as f(x)), which result from using a particular function.” For most linear

functions, the range could consist of any real number. If it is linear, the graph will most likely

extend infinitely, so any number might be used in the function. There are some situations when

y will equal a constant, that will give us a horizontal line, and in this situation the range would be

that one value.

Linear functions increase or decrease with a constant slope. These lines can have a

negative or positive slope which determines if the function will increase or decrease. Because of

the constant slope, the function will always be increasing or decreasing by the same common

difference.

Positive Linear Function Negative Linear Function


Exponential functions can also be increasing or decreasing, but

they are doing so at a common ratio instead of a common

difference. When an exponential function increases, it is known

as exponential growth and when it decreases it is known as

exponential decay. Using the formulaf ( x )=abx, we have

exponential growth when a > 0 and b is greater than 1. In exponential decay, a> 0 and b is

between 0 and 1.

Exponential Growth: Exponential Decay:

Concavity helps us determine how the graph of a function ‘bends’ at a certain point.

When we are talking about concave down, we use ‘concave’, and when we are talking about

concave up we use ‘convex’. In some functions there are times when the graph changes from

concave up to concave down, or the other way around. In this situation, when the graph changes


direction we call these points, inflection points. Linear and exponential functions do not have

any inflection points because they are continuous and do not change the direction in which they

are going at a single point.

There are two types of asymptotes that you could see in a function: vertical or horizontal.

The general definition of an asymptote can be defined as “a line, or curve, to which the graph

gets arbitrarily close. This means that for any distance named, no matter how small, the graph

will get within that distance and stay within that distance for some section of the graph with

infinite length.” In respect to linear and exponential functions, most asymptotes are not very

applicable to either. A linear function is a straight line that extends indefinitely; there is not a

point on the graph where the line gets arbitrarily close. In exponential functions, the graph is

continuing to increase at a common ratio, so there is rarely a point where the graph will not hit

when it is expanding. Most asymptotes are found in rational (ratio of integers) functions.

An intercept is “the points, if any, at which a graph crosses or touches the coordinate

axes”. Graphs can have a horizontal intercept, also known as the x-intercept, which is when the

function crosses or touches the x-axis, or a vertical intercept, also known as the y-intercept,

which is when the function crosses or touches the y-axis. In a linear function, you can find the

vertical intercept when using the equation f(x)= mx+b, “by evaluating the function when the

input variable, x, is 0 and is always the same as the constant b.” Using the same standard linear

equation, the “horizontal intercept is the value of the variable x when the function value is 0.”

(0=mx+b). There is only one y-intercept and one x-intercept for a linear function. In an

exponential function, you can find the vertical and horizontal intercepts by looking to see where

they cross the horizontal or vertical axis. When the x value is zero, we will have the y-intercept


of the function at that point. When the y-value is zero, we will find the x-intercept of the

function at that point.

There can be local and absolute, minimum and maximum values in a non-linear function.

In a linear function, there are no minimum or maximum values because it is one straight line, and

no open interval. In an exponential function, there are also no maximum or minimum values

because there are no true parabola’s in an exponential function, so there is no way to find the

maximum or minimum value.

A linear function is characterized by having a straight line associated with a common

difference between successive terms. When you start at a certain time on the highway and you

drive on cruise control at the same consistent speed for a certain amount of time, you would have

a linear line. Because the car is moving at a consistent speed, the time that it takes to drive a

mile would also be consistent. Using this information, we could figure out how long it would

take us to get a certain distance based off our speed per mile and travel time.

Exponential functions are known to start our growing slow, but then increase extremely

fast. They have a common ratio between successive terms. If you are a herpetologist and know

that a tree frog population doubles every week, and you are trying to get it out of extinction, you

could figure out how long it would take for the frog population to get to a certain number.

Exponential functions are helpful when dealing with terms that grow at an extremely fast rate.

“A rational function is a fraction of polynomials. That is, if p(x) and q(x) are polynomials, than

p (x)q(x )

is a rational function.” These functions are mainly represented in equation form, f(x)

(fraction of polynomials). These equation forms sometimes use words to describe the situation


that the equation represents. From using these equations we can set up a table to find our output

values, given our input values.

If we had the rational function: f(x)= x2+5x+2

X f(x)= x2+5x+2

-2 −22+5(−2 )+2

=

90=

Undefined

-1 −12+5−1+2

=

61=6

0 02+50+2

= 52

1 12+51+2

= 63=2

2 22+52+2

= 94

3 32+53+2

= 145

4 42+54+2

= 216

=72


From here, we have our x and y coordinates that we can plot on a graph. For the x value of -2

our function is undefined because when we plug -2 into the rational function, it gives us a 0 in

the denominator and that makes the function undefined.

The domain in a rational function is all the values that x is allowed to be (Any number

that would not make the denominator zero). Rational functions are restricted to only real number

values. For example: x+2

x2+2 x−15=0 is our function, we are only concerned with the denominator

because that is going to restrict the domain. So if we factor out our denominator we would get

(x+5)(x-3)=0, so if we set both of these to zero, we would have x+5=0 and x+3=0. When we

solve these equations, we have the zeroes of the denominator (-5 and 3). So our domain for

x+2x2+2 x−15

would be all real numbers except for -5 and 3, which would make the denominator

zero.

The easiest way to find the range in a rational function would be to look at a graph. By

graphing the function, it is easier to see what y-values are being represented and which are not.

With many rational functions there is an asymptote present (discussed below). The horizontal

asymptote can cause a break in our range values. The range in rational functions is restricted to

real number values.

Rational functions can increase or decrease depending on function. Here is an example

where the function is increasing and decreasing. f(x)= x

x−6


We can see that the line in the first quadrant is

increasing or has a positive slope, while the

line in the second and fourth quadrants are

decreasing with a negative slope. Most of these

functions are not increasing at a constant rate,

as we can see from this graph, these functions

are increasing exponentially. These functions

do not increase at a constant rate due to the

fraction of polynomials needed in order to

satisfy a rational function.

Another example of a rational function that is a little more complicated: f ( x )= x2−4x2−4 x

.

We would find our x-intercepts by setting the numerator to zero: x2−4=0, which would give us

our x-intercepts at 2 and -2. We can find our vertical asymptote by setting the denominator to

zero. x2−4x= x(x-4)= 0 which give us asymptotes at x= 0, and x=4. Here is a sketch of the

function. We can see that there are three distinctive sections that this function is broken up into

due to the asymptotes.


Asymptotes play a huge role in rational functions. It helps us to see where the function

approaches, but never touches. To find the vertical asymptote we use the domain of the function

to find which numbers are our zeroes (numbers that cannot be used in our domain because it

would make the denominator zero). At these points we will draw a vertical line, and at that point

we would have our vertical asymptote. Our rational function will climb up (increase) and slide

down (decrease) near this line, but never actually touch it. Every point where the denominator is

zero corresponds to a vertical asymptote, except when there is also a zero in the numerator. For

the horizontal asymptote, it really takes looking at a graph to be able to see it. At whatever point

the function approaches horizontally, that would be our horizontal asymptote for that function.

Like the vertical asymptote, the function will never touch the horizontal asymptote, but it will

continue to get closer and closer.

To find the y-intercept (vertical intercept) on the graph, we would set our x-values in the

function to zero to find where the function would cross the y-axis. To find the x-intercept

(horizontal intercept), we would set the numerator equal to zero and then solve for x. For

example if we were given the function 1

x2+x−2 we see that there are no x’s in the numerator, so

this rational function would have no x-intercepts, but when we substitute 0’s for all the x’s than

we would have 1

(0)2+(0)−2 which would equal ½. So our function would cross the y-intercept

at (0, ½ ).

There are big companies that want to predict how their sales are going to be in the future

from the release of a new product. So the weekly sale S (in thousands of units) for the ‘t’th week

after the introduction of the product in the market is given by S= 120 t

t2+100. Through this rational


function we could see how many sales the product is predicted to sell by a certain week. This

has proven resourceful to many companies in helping them figure out their future business plans.

By seeing their sales, they can see how many of a product they should be making, or if they

should slow down production because the item is not selling how they intended.

From this assignment, I have gained a deeper knowledge of functions: the different types,

their different characteristics, and their importance in real world situations. Prior to researching

functions, I knew the basics of a few types of functions but nothing in depth. Researching linear

and exponential functions gave me a more full understanding of how these functions work and

what they are made of. Since these functions are some of the most common functions used,

knowing them in detail will be beneficial when understanding how other functions work when

they have similar or different characteristics. After researching, my basic definition of a function

could be described as a relationship between two groups of data where every input value has one

output value. This definition could be used for any type of function, but I think it is seen most

clearly in linear functions. I found it interesting that this definition proved true for all types of

functions even though they all had very different shapes, equations and visual characteristics.

By understanding exponential and linear functions in depth has helped me understand how I

want to teach and portray this information later in my career when I am a teacher. It will allow

me to be able to teach my students in confidence and be able to use other information related to

certain types of functions to explain a certain topic.


References

Concavity and Points of Inflection. Concavity and Points of Inflection. Retrieved April 24, 2014, from http://www.sosmath.com/calculus/diff/der15/der15.html

Dawkins, P. Pauls Online Notes : Algebra - Rational Functions. Pauls Online Notes : Algebra - Rational Functions. Retrieved April 26, 2014, http://tutorial.math.lamar.edu/Classes/Alg/GraphRationalFcns.aspx

"Finding Domain and Range." Finding Domain and Range. Monterey Institute. Web. 14 Apr. 2014. <http://www.montereyinstitute.org/courses

Foerster, P. A. (2005). Calculus: Concepts and applications. Berkeley, CA: Key Curriculum Press.

Horizontal and Vertical Intercepts. Earth Math. Retrieved April 24, 2014, from http://earthmath.kennesaw.edu/main_site/review_topics/intercepts.htm

Katz, V. J. (2009). A history of mathematics: an introduction (3rd ed.). Boston: Addison-Wesley.

Larson, R., Hostetler, R. P., & Falvo, D. C. (2007). Precalculus with limits. Boston, MA: Houghton Mifflin.

Martin, O. Math tutorial: concave and convex functions of a single variable.Math tutorial: concave and convex functions of a single variable. Retrieved April 24, 2014, from http://www.economics.utoronto.ca/osborne/MathTutorial/CV1F.HTM

Math Glossary. Math Glossary. Retrieved April 24, 2014, from http://www2.seminolestate.edu/srickman/glossary/asymptote.htm

freeMATHhelp. Domain and Range. Retrieved April 24, 2014, from http://www.freemathhelp.com/domain-range.html

Motz, L., & Weaver, J. H. (1993). The story of mathematics. New York: Plenum Press.

Ponte, J. P. (n.d.). The History of the Concept of Function and Some Educational Implications. The Program of Mathematics: University of Georgia. Retrieved March 30, 2014, from http://math.coe.uga.edu/TME/Issues/v03n2/Ponte.pdf

"Rational Functions." Rational Functions. Web. 12 Apr. 2014. <http://math.ucsd.edu/~wgarner/math4c/t

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hendersonfinalportfolio.weebly.comhendersonfinalportfolio.weebly.com/uploads/1/7/6/8/1768… · Web viewA pre-calculus textbook stated that “A function f from a set A to a set