Chapter 4: Algebraic Fractions Lesson 3 Addition and Subtraction UNIT TEST # 1 - Jan 24.
MAT01A1: Functions and Mathematical Models · An algebraic function is one that can be formed by...
Transcript of 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.