mat01a1: functions and mathematical models · an algebraic function is one that can be formed by...

MAT01A1: Functions and Mathematical Models

Dr Craig

4/5 March 2020

Reminder: HW2 on WeBWorK

I On complex numbers

I Closes Friday at 23h59

Saturday class

I D-LES 101

I 09h00 – 12h00

I Revision to help prepare for Monday’s test

Semester Test 1

I Full details in the pdf posted under

“Assessments”.

I The venue allocation is very important:

Surnames Arends – Moses:D1 LAB K08

Surnames Motaung – Zwane:D-LES 101

I You must be at your venue by 08h15.

I Test is 08h30–10h00.

Consultation Hours

Thursday:

08h50 – 10h25 Dr Robinson (C-Ring 514)

11h20 – 12h15 Dr Craig (C-Ring 508)

15h00 – 16h15 Dr Craig (C-Ring 508)

Maths Learning Centre (C-Ring 512)

Wednesday 08h00 – 15h25

Friday 08h00 – 15h25

Introduction to functions

Four examples of functions:

I The area of a circle depends on the

radius: A = πr2.

I Population of the world depends on time:

P (1950) = 2, 560, 000, 000.

I The cost of posting a package depends on

the weight: C(w).

I The vertical ground acceleration during

an earthquake: S(t).

Example:

A rectangular box with an open top has a

volume of 10m3. The length of the base is

twice its width. Material for the base costs

R10/m2 and material for the sides costs

R6/m2. Express the cost of the materials as

a function of the width of the base.

Solution:

C(w) = 20w2 +180

w, w > 0

Domain and range of functions

A function f is a rule which assigns to each

element x in a set D exactly one element,

f (x), in a set E.

The set D is the domain of f . The range of

f is the set of all possible values of f (x) as

x varies through the domain.

A symbol representing an arbitrary element of

the domain is called an independent variable

and a symbol representing an arbitrary

element of the range is a dependent variable.

In the example of the circle: r is the

independent variable while A is the

dependent variable.

Graphs of functions: a common way of

representing a function is by a graph.

Formally, the graph of the function f is the

set of ordered pairs {(x, f (x)

)| x ∈ D }.

7

1

4

1 4 7

Dom(f)= [1,6] and Ran(f)=[1,7]

Examples

Sketch the following functions and find their

domain and range:

(a) f (x) = −3x + 4

(b) g(x) = x2 − 2

(c) h(x) = cscx

Representations of functions

There are four ways to represent a function:

I Verbally (describe in words)

I Tables

I Graphically

I Algebraically

The vertical line test

How do we know if a curve is a function?

A curve in the xy-plane is a function of x if

and only if no vertical line intersects the

curve more than once.

The curve x2 + y2 = 4 is not a function:

The curve y =√4− x2 is a function:

Piecewise defined functions

We are already familiar with one example of

a piecewise defined function, the absolute

value function:

f (x) = |x| =

{x if x > 0

−x if x < 0

Now sketch the function

f (x) =

{−x + 2 if x 6 1

x2 if x > 1

Example: step functions

Example: step functions

Consider the cost of travelling on the Rea

Vaya bus. Here x is the number of km

travelled and C(x) is in rands:

C(x) =

8 if 0 6 x 6 5

9.60 if 5 < x 6 10

11.80 if 10 < x 6 15

13.90 if 15 < x 6 25

15 if 25 < x 6 35

16 if 35 < x

Properties of functions

Symmetry in functions

If a function f satisfies

f (−x) = f (x) for all x ∈ D

then f is an even function.

Examples:

I f (x) = x2

I f (x) = cos(x)

I f (x) = |x|Another way of defining an even function is

to say that it is a reflection about the y-axis.

Symmetry in functions

A function is odd if

f (−x) = −f (x) for all x ∈ D

Examples:

I f (x) = x

I f (x) = sin(x)

I f (x) = x3

An odd function is a reflection about the

origin. A necessary condition for a function f

to be odd is that it must have f (0) = 0.

Increasing and decreasing functions

A function f is increasing on an interval I if

whenever x1 < x2, we have f (x1) < f (x2).

A function f is decreasing on an interval I if

whenever x1 < x2 we have f (x1) > f (x2).

Example: Is the function f (x) = cosx

increasing, decreasing, or neither over the

following intervals:

(a) x ∈ [π, 3π/2]

(b) x ∈ [0, π/2]

(c) x ∈ [π/2, 3π/2]

Ch 1.2: a catalogue of essentialfunctions

Mathematical models

A mathematical model is a mathematical

description, using a function or an equation,

of a real-world problem.

If y is a linear function of x then the graph is

a straight line:

y = mx + c

where m is the slope of the graph, and c is

the y-intercept.

Example of a linear model: The length

of a newly born snake is 10cm and after 3

months the length is 25cm. It grows the

same amount each month.

(a) Express length as a function of time

(in months) where the D = [0, 12].

(b) Draw the graph of the length

function.

(c) What is the length of the snake

after 9 months?

Functions from data: If we don’t have a functionto work from, we can try to determine a functionusing empirical data.

Year Number of registered cars in SA

2005 4,500,000

2006 4,670,000

2007 4,890,000

2008 5,100,000

2009 5,310,000

2010 5,540,000

2011 5,770,000

2012 5,900,000

For the last example we can try to find a

function which fits the data by calculating

the slope of the line which goes through the

first and last points.

m =5, 900, 000− 4, 500, 000

2012− 2005

Therefore m = 200, 000.

To be more accurate, we can use a statistical

technique known as linear regression.

More types of functions (mostlynon-linear)

Polynomials

These are functions of the form:

f (x) = anxn+an−1x

n−1+. . .+a2x2+a1x+a0

where a0, . . . , an are constants. If the

leading coefficient an 6= 0 then f (x) is a

polynomial of degree n.

I polynomial of degree 1 = linear function

I polynomial of degree 2 = quadratic

function

I polynomial of degree 3 = cubic function

Power functions

These are functions of the form

f (x) = xa

Note that a can be

I a positive integer

I a = 1n where n is a positive integer

I a negative integer, so f (x) = 1xa

Rational functions

A rational function f is a ratio of two

polynomials

f (x) =P (x)

Q(x)

Example:

f (x) =2x4 − x2 + 1

x2 − 4

Algebraic functions

An algebraic function is one that can be

formed by using the algebraic operations of

addition, subtraction, multiplication, powers,

division and taking roots.

Note: any rational function is automatically

an algebraic function.

Examples:

f (x) =√x2 + 1 h(x) =

x4 − 16x2

x +√x

Trigonometric functions

Functions that express the ratio between x, y

and r when angles are plotted on the

xy-plane.

I sinx

I cosx

I tanx

I cscx

I secx

I cotx

Exponential functions

These are functions of the form

f (x) = ax

where a > 0 is constant.

Examples:

I f (x) = 2x

I f (x) = 0.5x

I f (x) = ex

f (x) = 2x

f (x) = 0.5x

Logarithmic functions

f (x) = loga x where a > 0 is a constant

(also known as the base). Logarithmic

functions are the inverse of exponential

functions. That is,

if y = ax then loga y = x.

f(x) = log2 x

Make sure that you are comfortablewith the sketches of exponential andlogarithmic functions. Know theirshape and where the intercepts occur.