functions ii.functions whose graphs are horizontal lines are the constant functions. note that...

Szent Istvan University, Faculty of Veterinary ScienceDepartment of Biomathematics and Informatics

Biomathematics 4

Functions II.Janos Fodor

Copyright c© [email protected] Revision Date: September 11, 2006 Version 1.25

mailto:[email protected]

Table of Contents

1 Important function classes 3

1.1 Linear functions . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Power functions . . . . . . . . . . . . . . . . . . . . . . . . . 5

• Application of Power Functions: How big can a cell be? . . . 7

1.3 Polynomial and Rational Functions . . . . . . . . . . . . . . . 10

1.4 Exponential Functions . . . . . . . . . . . . . . . . . . . . . . 12

• The Exponential Function with Base e . . . . . . . . . . . . 14

1.5 Logarithmic Functions . . . . . . . . . . . . . . . . . . . . . . 15

1.6 Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . . 17

• The Sine Function . . . . . . . . . . . . . . . . . . . . . . 18

• The Cosine Function . . . . . . . . . . . . . . . . . . . . . 19

• The Tangent Function . . . . . . . . . . . . . . . . . . . . 20

• The Cotangent Function . . . . . . . . . . . . . . . . . . . 20

Section 1: Important function classes 3

1. Important function classes

We consider some important particular classes of functions, namely:

• linear functions,

• power functions,

• polynomials and rational functions,

• exponential functions,

• logarithmic functions,

• periodic functions.

1.1. Linear functions

Linear FunctionA function f is called linear if it can be written in the form:

f(x) = mx + b,

where m and b are real numbers.

Example. The pressure y measured x meters below the sea level is y = 0.1x+1atmosphere. Thus, y is a linear function of x with m = 0.1, b = 1.

The graph of a linear function is a line. If m = 0 then f(x) = b, and the graphof f is a horizontal line.

Functions whose graphs are horizontal lines are the constant functions.

Note that vertical lines are not graphs of functions. Vertical lines always havethe form x = c, where c is some constant (x = 3, for example).

The domain of a linear function is always the set of all real numbers R. Ifa linear function is not constant, its range is also R. If it is constant, e.g.,f(x) = b, then Rf = {b}.


The constants b and m in the form f(x) = mx+b give us important informationabout the line we wish to graph.

We can see, by choosing x = 0, that the line passes through the y-axis at thepoint (0, b). In other words: the y-intercept of f is (0, b) (we simply say they-intercept of f is b).

The second constant m, which is the coefficient of x, tells us the steepness orslope of the line.

Let P1(x1, y1) and P2(x2, y2) be points on a line. Then

Slope =vertical change (rise)

horizontal change (run)=

y2 − y1

x2 − x1.

If the horizontal change is 0, then the line is vertical and has no slope. If thevertical change is 0, then the line is horizontal and has zero slope.

Geometric Interpretation of Slope


How can we tell when two nonvertical lines are parallel or perpendicular to eachother? The following theorem provides a convenient test.

Parallel and Perpendicular LinesLet f1(x) = m1x + b1, f2(x) = m2x + b2 be linear functions. Their graphsare

• parallel if and only if m1 = m2 ;

• perpendicular if and only if m1 ·m2 = −1 .

1.2. Power functions

Power FunctionA function f is called a power function if it can be written in the form

f(x) = xp,

where p is any real number.

For example, f(x) = x2, g(x) = x−3, h(x) = x1/2 are all power functions.

So are y =1

x2 and y = 3√

x since they can be rewritten as y = x−2 and

y = x1/3, respectively.

Note that sometimes functions f(x) = c · xn are also called power functions(where c 6= 0 constant).

Consider two well-known examples now.

2x3 and 3x2


When n is a positive integer, we see in the next figure that power functions arerelatively shallow near the origin, but go steeply beyond |x| = 1, i.e. for x

larger than 1 or smaller than −1.

For larger values of the power n, the graph of the power function y = xn getsflatter close to the origin and steeper for |x| > 1.

Symmetry properties of power functions depend on whether n is even (thegraphs are all symmetric about the y axis; a function with this property is calledan even function) or n is odd (the graphs are all symmetric about the origin;a function with this type of symmetry is called an odd function).

In the next figure we can see graphs of a few of the even (y = x2; y = x4;y = x6 ) and odd (y = x; y = x3; y = x5 ) power functions.

Power functions with integer powers satisfy an important relationship with re-spect to one another:

For larger powers, the function y = xn, gets flatter (and smaller) close tox = 0 and steeper (and larger) for large values of x.

For example, at x = 0.1, the function y = f(x) = x2 takes on a larger value


(f(0.1) = 0.01) than the function y = g(x) = x4 (g(0.1) = 0.0001). At x = 2,the roles are reversed. (f(2) = 4, whereas g(2) = 16.)

We say that the low powers dominate close to x = 0, while the higher powersdominate for large x. This will have important implications on the relativeeffects of terms of various powers in a polynomial.

• Application of Power Functions: How big can a cell be?

Most applications of power functions in biology are related to processes of thesurface or the volume of organisms.

Now we try to answer the following questions:

1. What determines the size of a cell and why some size limitations exist?

2. Why should animals be made of millions of tiny cells, instead of just a fewhundred large ones?

While these questions seem extremely complicated, a relatively simple mathe-matical argument can help in finding the answers.

We will formulate a mathematical model.

A model is just a representation of a real situation which simplifies things byrepresenting the most important aspects, while neglecting or idealizing theother aspects.

Our model is based on the following assumptions:

1. The cell is spherical.

2. The cell absorbs oxygen and nutrients from the environment through itssurface. We will assume that the rate at which nutrients (or oxygen) areabsorbed is proportional to the surface area, S, of the cell.

3. The rate at which nutrients are consumed in metabolism (i.e. used up) isproportional to the volume, V , of the cell.


Model of a single cell.

We define the following quantities relevant to a single cell:

A = net rate of absorption of nutrients per unit time.

C = net rate of consumption of nutrients per unit time.

V = cell volume.

S = cell surface area.

r = radius of the cell.

We now rephrase the assumptions mathematically.

By assumption (2), A is proportional to S. This means that

A = k1S ,

where k1 > 0 is a constant of proportionality.

The value of this constant would depend on the permeability of the cell mem-brane, how many pores or channels it contains, and/or any active transportmechanisms that help transfer substances across the cell surface into its inte-rior.

By assumption (3), C is proportional to V , so that

C = k2V ,

where k2 > 0 is a second proportionality constant. The value of k2 woulddepend on the rate of metabolism of the cell, i.e. how quickly it consumesnutrients in carrying out its activities.

By the first assumption, the surface area and volume of the cell are:

V =4

3πr3 , S = 4πr2 .


Putting these facts together leads to the following relationships between nutrientabsorption, consumption, and cell radius:

A = k1(4πr2) = (4πk1)r2 ,

C = k2

(4

3πr3

)=

(4

3πk2

)r3.

Note that A and C are now quantities that depend on the radius of the cell.

In order for the cell to survive, the overall rate of consumption of nutrientsshould balance (be equal to) the overall rate of absorption, i.e. C = A:(

4

3πk2

)r3 = (4πk1)r

2 .

One solution is r = 0 (not interesting). If r 6= 0, then we can cancel a factor ofr2 from both sides to obtain the value of the radius r at which nutrient balanceoccurs:

r = 3k1

k2.

We know: for large values of r, higher powers dominate; for small r lowerpowers dominate.

Since at r = 3k1

k2the two functions are equal, it follows that

• for smaller cell sizes the absorption A ≈ r2 is the dominant process;

• for large cells, the consumption C ≈ r3 is higher than absorption.

We conclude:

Cells larger than the critical size r = 3k1

k2will be unable to keep up with the

nutrient demand, and will not survive.

Thus, the size of the cell has strong implications on its ability to absorb oxygenand nutrients quickly enough to feed itself. For these reasons, cells larger thansome maximal size (roughly 1 mm in diameter) rarely occur.


Furthermore, organisms that are bigger than this size cannot rely on simplediffusion to carry oxygen to their parts—they must develop a circulatory systemto allow more rapid dispersal of such life-giving substances or else they willperish.

Further example:

Example.

For plants and animals with a shape that is more complicated than the sphere,there is still an easy geometric relation between the volume and the surface ofthe body. The volume is a cubic and the surface is a quadratic function ofthe linear dimension (such as length, height) of the body. Therefore, the sizeof the animals can only vary within a certain interval.

As an illustration, take a giant mouse, in shape like an ordinary one but witha linear dimension (e.g. length) ten times greater. From the above facts itfollows that the mass of the giant mouse is a thousand times greater than anordinary one, while the surface of its lung is only a hundred times greater.This mouse can hardly survive.

1.3. Polynomial and Rational Functions

Linear (y = mx + b) and quadratic functions (y = ax2 + bx + c) are specialtypes of more general mathematical functions called polynomial functions orsimply polynomials. Examples of polynomial functions:

f(x) = 5x3 + 2x2 − 3x + 5, g(x) = x2 − 1, h(x) = 6.

Note that polynomials of degree higher than 2 are occasionally used in biology.

Polynomial FunctionA function f is a polynomial function of degree n if

f(x) = anxn + an−1x

n−1 + . . . + a1x + a0,

where a0, a1, . . . , an are real numbers, n is a nonnegative integer, and an 6= 0.

Example.

(a) f(x) = x is a polynomial.


(b) f(x) = 1x is not a polynomial since 1

x = x−1, and −1 is negative.

(c) f(x) = 16x is a polynomial (n = 1, an = 1

6).

(d) f(x) =√

x + 3x2 is not a polynomial since√

x = x1/2 and 1/2 is not aninteger.

We postpone the graphing of polynomial functions until we discuss curve sketch-ing as application of differential calculus later on. The following figure is onlyan illustration.

Three polynomials of degree 3.

Behaviour for large x: All polynomials are unbounded as x → +∞ and asx → −∞. In fact, for large enough values of x, we have seen that the powerfunction y = f(x) = xn with the largest power, n, dominates over other powerfunctions with smaller powers. For

f(x) = anxn + an−1x

n−1 + . . . + a1x + a0

the highest power term will dominate for large x. Thus for large x (whetherpositive or negative)

f(x) ≈ anxn (whenever |x| is large).

Behaviour for small x: Close to the origin, we have seen that power functionswith smallest powers dominate. This means that for x ≈ 0 the polynomial isgoverned by the behaviour of the smallest (non-zero coefficient) power, i.e,

f(x) ≈ a1x + a0 (for small x).


Now we define rational functions (remember the definition of rational numbers).

Rational FunctionsA function is called rational if it is the quotient of two polynomial functions.

For example, the function

f(x) =x2 − 1

x2 − 3x + 2is a rational function. You will have the chance to meet more rational functionsin later subjects.

1.4. Exponential Functions

Exponential functions can be used as models for certain types of growth ordecay.

Example: Growth of a foal.

We have a foal with weight 50 kg. The weight increases at a rate of 20%

during consecutive time intervals of equal length. Then the weights at the end

of 0, 1, 2, . . . time intervals are

50, 50

(1 +

20

100

), 50

(1 +

20

100

)2

, 50

(1 +

20

100

)3

, . . .

In general, if the initial weight is c and the rate of growth is p then the weights

at the end of 0, 1, 2, . . . time intervals are

c, c(1 +

p

100

), c

(1 +

p

100

)2

, c(1 +

p

100

)3

, . . .

If b := 1 + p100 then the weight after x time interval is

c · bx (x = 0, 1, 2, . . .).

An animal does not grow in steps, it grows continuously. Does the previousexpression have any meaning if x is a real number?

Mathematically speaking, we try to replace the domain {0, 1, 2, . . .} by the setR of all real numbers.

At this stage you should take it for granted that this is possible. The corre-sponding function is called exponential.


Exponential Function The equation

f(x) = bx (b > 0, b 6= 1)

defines an exponential function for each different constant b, called thebase. The independent variable x may assume any real value.

Thus, the domain of f is the set of all real numbers, and it can be shown thatthe range of f is the set of all positive real numbers.

Sometimes functions defined by

f(x) = c · ax,

where a 6= 1 is a positive real number, c 6= 0 is a real number, are also calledexponential functions.

It is useful to compare the graphs of y = 2x and y = (1/2)x = 2−x by plottingboth on the same coordinate system.

The graph of f(x) = bx for b > 1 looks very much like the graph of theparticular case y = 2x, and the graph of f(x) = bx for 0 < b < 1 looks verymuch like the graph of y = (1/2)x.

Basic Properties of the Graph of f(x) = bx, b > 0, b 6= 1

1. All graphs pass through the point (0, 1).

2. All graphs are continuous, with no holes or jumps.

3. The x axis is a horizontal asymptote.

4. If b > 1, then bx increases as x increases.

5. If 0 < b < 1, then bx decreases as x increases.

6. The function f is one-to-one.


• The Exponential Function with Base e

The following expression is important to the study of calculus (m is a positiveinteger): (

1 +1

m

)m

.

Interestingly, by calculating the value of the expression for larger and largervalues of m (see Table below), it appears that [1 + (1/m)]m approaches anumber close to 2.7183.

It can be shown that as m “increases without bound”, the value of [1+(1/m)]m

approaches an irrational number that we call e. Just as irrational numbers suchas π and

√2 have unending, nonrepeating decimal representations (see Chapter

1), e also has an unending, nonrepeating decimal representation. To 12 decimalplaces,


The constant e turns out to be an ideal base for an exponential function. Thisis why you will see e used extensively in expressions and formulas that modelreal-world phenomena.

Since e is a positive number different from 1, for any real number x the equa-tion f(x) = ex defines the exponential function with base e. The exponentialfunction with base e is used so frequently that it is often referred to as theexponential function. The graphs of y = ex and y = e−x are shown in the nextfigure.

1.5. Logarithmic Functions

Consider again the example of growth of a foal. We can ask: at what time ofgrowth does the weight reach 86.4 kg?

That is, the value of y (the dependent variable) is given; find the correspondingvalue of the independent variable x:

50 · 1.2x = 86.4.

A new class of functions is required, called logarithmic functions, as inversesof exponential functions.

If we start with the exponential function, f defined by y = 2x and interchangethe variables x and y, we obtain the inverse of f , denoted by f−1 and definedby the equation x = 2y.

The graphs of f , f−1, and the line y = x are shown in the next figure. Thisnew function is given the name logarithmic function with base 2.


Since we cannot solve the equation x = 2y for y using the algebraic propertiesdiscussed so far, we introduce a new symbol to represent this inverse function:

y = log2 x (read: “log to the base 2 of x”).

Thus,

y = log2 x is equivalent to x = 2y.

In general, we define the logarithmic function with base b to be the inverse ofthe exponential function with base b (b > 0, b 6= 1), where

y = logb x is equivalent to x = by.

Logarithmic FunctionThe function f defined for x > 0 by

f(x) := logb x,

where b > 0, b 6= 1, is called the logarithmic function with base b.

The domain of a logarithmic function is the set of all positive real numbersand its range is the set of all real numbers. Thus, log10 3 is defined, but log10 0


and log10(−5) are not defined. Typical logarithmic curves are shown in the nextfigure.

Properties of Logarithmic Functions

If b, M , N are positive real numbers, b 6= 1, and p and x are real numbers, then:

1. logb 1 = 0 5. logb MN = logb M + logb N

2. logb b = 1 6. logbMN = logb M − logb N

3. logb(bx) = x 7. logb(Mp) = p logb M

4. blogb x = x (x > 0) 8. logb M = logb N if and only if M = N .

Common and Natural Logarithms

Of all possible logarithmic bases, the base e and the base 10 are used almostexclusively.

Common logarithms: logarithms with base 10.

Natural logarithms: logarithms with base e.

1.6. Periodic Functions

Take a look at the following graph, which shows the approximate average dailyhigh temperature in New York’s Central Park:


Each year, the pattern repeats over and over, resulting in the following graph.

This is an example of cyclical or periodic behavior.

Periodic functions describe processes that have parameters of dynamics repeat-ing in time. For instance, the period of the average daily temperature is 1 yearand the period of the parameters of the heart functions is a fraction of onesecond.

A function f is called periodic if there exists a positive number T such that

f(x + T ) = f(x)

for all real number x. The smallest T with this property is called the periodof f .

We model cyclical behavior using the sine, the cosine, the tangent and thecotangent functions.

• The Sine Function

The sine of a real number t is given by the y-coordinate (height) of the pointP in the following diagram, in which t is the distance of the arc shown.

sin(t) = y-coordinate of the point P


The period of sin is 2π. The graph of sin:

• The Cosine Function

cos t = the x-coordinate of the point P shown

Fundamental Trigonometric Identity:sin2 t + cos2 t = 1.

The period of cos is 2π. The graph of cos, as you might expect, is almostidentical to that of the sine function, except for a ”phase shift” (see the figure).


• The Tangent Function

tan x :=sin x

cos x. The period of tan is π.

• The Cotangent Function

cot x :=cos x

sin x. The period of cot is π.

functions ii.functions whose graphs are horizontal lines are the constant functions. note that...

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