Analysis I

Functions and their Representations - Pesonen 2002-2016

- What a function is?
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About this Worksheet
The purpose of this Worksheet is to reinforce the understanding of the concept "function" of one variable. Especially,
- the role of the domain (set) and the co-domain (set), here usually called goal set
- various representations of the rule that relates these sets.

Contents

I Definition of function
II Representations of function
III Vector valued function of a real variable
IV Student Feedback

I Definition of function: domain - rule - goal set

A. Definition of function

Let X and Y be non-empty sets.
A function f (also mapping) from the set X into the set Y is a rule (given in some way or agreed to exist), which associates to every element of X a uniquely determined element of Y.
The set X is called the domain and Y the co-domain or the goal set.

The image of a subset A of X is the set

f(A) = {f(x) | x in A}.
The image of the whole domain is called the range of the function f.

The pre-image of a subset B of Y is the set

f -1(B) = {x in A | f(x) in B}.

The rule specific to the function can be, for example

figure or diagram,
verbal expression,
relation catalogue of pairs (x,f(x)),
table of values,
defining the values using an expession or formula, directly or implicitly by an equation.

B. Ways to represent a function

In the following figures the pushing of button RULE shows the rule as a graphical animation.
Domain-goal set diagram
Example
This page requires a Java capable browser. Beside we can have a graphical representation of a rule relating the sets X and Y, which defines a function from X into Y. The rule can be represented in symbolical form:
f(n) = n + 1,
verbally:
"We add 1 to the variable (element of the domain).",
as a catalogue:
f(1) = 2, f(2) = 3, f(3) = 4
and as a subset of the cartesian product set X×Y:
f = {(1,2), (2,3), (3,4)}.
Domain-goal set diagram
Problem 1
Beside we can see a rule from X into Y.
This page requires a Java capable browser.
a) Is this a function from X into Y? yes no don't know
b) If it is not a function, explain why:
c) Express the rule symbolically.
f(n) =
d) Express the rule verbally.
Domain-goal set diagram
Problem 2
Beside we can see a rule from X into Y.
This page requires a Java capable browser.
a) Is this a function from X into Y? yes no don't know
b) If it is not a function, explain why:
c) Express the rule symbolically.
f(n) =
d) Express the rule verbally.
Domain-goal set diagram
Problem 3
Beside we can see a rule from X into Y.
This page requires a Java capable browser.
a) Is this a function from X into Y? yes no don't know
b) If it is not a function, explain why:
c) Express the rule in catalogue form.
d) Express the rule as a subset of X×Y.
Domain-goal set diagram
Problem 4
Beside we can see a rule from X into Y.
This page requires a Java capable browser.
a) Is this a function from X into Y? yes no don't know
b) If it is not a function, explain why:
c) Express the rule in catalogue form.
d) Express the rule as a subset of X×Y.

C. Bijection

A function is called bijection if it has the following properties
  1. no two domain elements are mapped to a same element of the goal set (injectivity),
  2. to every goal set element at least one domain element is mapped (surjectivity).
In the following figure you see the rule by ANIMATE button or moving the variable x by the computer mouse.
Domain and goal set on axis
Problem 5
Beside we can see a rule from X into Y.
This page requires a Java capable browser.
a) Is this a function from X into Y? yes no don't know
b) If it is not a function, explain why:
c) Express the rule symbolically.
f(n) =
d) How can you, by suitable limitations, obtain a bijection?

II Representations of function

Let us observe a one real variable function f: R ® R on an interval.
The following dynamical figures show how a real variable x and its real image f(x) can be presented in the light of the definition and, on the other hand, as a graph or curve in plane coordinate system.
Both have their own good properties.

A. One-dimensional visualisation: domain-rule-goal set

In the following figures there are many kinds of control buttons and tools.
In general, at least the variable x can be dragged by the mouse.
The Trace points can be wiped by the button x on the right lower corner.
All are worth exploring freely!
Real functions: axis-axis representation
Problem 6
Beside is descibed a real variable function which we investigate on the interval visible in the picture.
This page requires a Java capable browser.
a) What is the value of the function at the point 2 ?
b) At which point(s) the function has the value 1 ?
c) What is the minimum of the function ?
d) What is the maximum of the function ?
e) What values the function gets, i.e. what is its range ?
f) What is the image of the interval [0,1] ?

Real functions: axis-axis representation
Problem 7
Beside is descibed a real variable function which we investigate on the interval visible in the picture, but we also may try to predict.
This page requires a Java capable browser.
a) What is the value of the function at the point 2 ?
b) Is this function continuous ? yes no don't know
c) What is the limit of the function at 2 when approaching from the right?
d) What is the limit of the function at 2 when approaching from the left?
e) What seems to be the limit of the function at infinity ?
Real functions: axis-axis representation
Problem 8
Beside is described a real function defined on an interval.
This page requires a Java capable browser.
a) What is the domain of the function ?
b) What is the range of the function ?
c) Is the function monotone, i.e. increasing in its whole domain or decreasing in its whole domain ? yes no don't know
d) Explain your previous answer!
e) Is this function continuous ? yes no don't know
Real functions: axis-axis representation
Problem 9
Beside is described a real function defined on an interval.
This page requires a Java capable browser.
a) What is the domain of the function ?
b) What is the range of the function ?
c) Where is this function increasing, where decreasing ?
d) Is it a bijection, if the range is taken to be its goal set ? yes no don't know
Real functions: axis-axis representation
Problem 10
Beside a situation concerning a real variable x is concerned.
This page requires a Java capable browser.
a) Is this a real valued function of a real variable ? yes no don't know
b) Explain your previous answer!
Real functions: axis-axis representation
Problem 11
Beside a situation concerning a real variable x is concerned.
This page requires a Java capable browser.
a) Is this a real valued function of a real variable ? yes no don't know
b) Explain your previous answer!

B. Representing by Plane Graphs

It should be clear, that in many respects the one-dimensional visualisation provides us with a poor analysing tool.
Therefore, one variable functions are often dealt with as relations, roughly speaking as ordered pairs (x,f(x)) in two-dimensional coordinate system.
Real functions: coordinate system
Problem 13
Beside a situation concerning a real variable x is concerned.
This page requires a Java capable browser.

Push the button "TO COORDINATE SYSTEM" and find (with the help of the representation) which of the following properties the function has:

a) The function is increasing on the interval [1,2]. yes no don't know
b) The function has a maximum on the interval [4,5]. yes no don't know
c) The function is continuous on the visible interval. yes no don't know
d) The function is peridic. yes no don't know
e) The function is differentiable on the interval [0,1]. yes no don't know
f) The derivative of the function at 1 is negative. yes no don't know
g) There are points of non-differentiability. yes no don't know
h) The function has points of inflection. yes no don't know
Real functions: values and graph
Problem 14
Beside a situation concerning a real variable x is concerned in the coordinate system.
This page requires a Java capable browser.

Which of the following statements are true:

a) The function is increasing on the interval [1,2]. yes no don't know
b) The function has a maximum on the interval [4,5]. yes no don't know
c) The function is continuous on the visible interval. yes no don't know
d) The function is peridic. yes no don't know
e) The function is differentiable on the interval [0,1]. yes no don't know
f) The derivative of the function at 1 is negative. yes no don't know
g) There are points of non-differentiability. yes no don't know
8) The function has points of inflection. yes no don't know
Real functions: values and graph
Problem 15
Beside a situation concerning a real variable x is concerned in the coordinate system.
This page requires a Java capable browser.
a) What is the value of the function at the point 2 ?
b) Is this function continuous? yes no don't know
c) What is the limit of the function at 2 when approaching from the right?
d) What is the limit of the function at 2 when approaching from the left?
e) What seems to be the limit of the function at infinity ?

C. Examples

Now we examine some elementary functions.
Real functions: power and root xr
Problem 16
Beside we describe power and root functions.
This page requires a Java capable browser.

Find out the following:

a) For which values r the curve y = xr is increasing ?
b) For which values r the curve y = xr goes through the point (3,2) ?
c) For which values r the curves y = xr and y = x1/r are the same ?
d) For which values r the curves y = xr and x1/r are reflections of each other with respect to the line y = x ?

Real functions: exponential functions ax
Problem 17
Beside we describe exponential functions.
This page requires a Java capable browser. The base value a can be controlled on the parameter axis in the bottom, where also the Neper number e is marked.
a) For which values a the curve y = ax is increasing ?
b) For which values a the curve y = ax goes through the point (3,2) ?
c) For which values a the curve y = ax is the reflection of the curve y = ex with respect to the vertical axis ?

Real functions: Logarithms logax
Problem 18
The inverse function of the exponential function x -> ax of general base a is the logarithm of base a, in which each strictly positive number x is mapped to the real number y = logax satisfying ay = x.
Especially ln = loge.
This page requires a Java capable browser. With buttons you can have visible the general a-based logarithm loga. For the sake of comparison, you also see the separately fixed the natural logarithm x -> ln x.
The base value a can be controlled on the parameter axis in the bottom, where also the Neper number e is marked; there is also the Briggs base 10 (how can you see it?!).
a) For what values a and x the logarithm logax is defined?
b) For what values a the function y = logax is decreasing ?
c) For what values a the curve y = logax goes through (2,-1) ?
d) For what values a the curves y = ln x and y = logax are reflections of each other (with respect to a suitable line) ? What is the exact value of a then ?
e) For what values a the curve y = logax turns vertical ?

III Vector valued function of a real variable

When two real-valued functions are placed to form a vector, we have a vector-valued function of one real variable, a two-dimensional vector function.
When two continuous functions are mapped to two separate coordinate axis, we get a plane curve.
Vector functions suit well to describing e.g. multidimensional movement.
Two real functions together

If f and g are real-valued functions, a vector function is born:

H(x) = (f(x), g(x)).
Problem 19
Beside we describe a situation where to each real number x there corresponds an image vector (f(x), g(x)).
This page requires a Java capable browser. Transform to the coordinate system by pushing "TO PLANE GRAPH" and try the different buttons. Reset when needed.
Tell the following things about the system:
a) What is the value of the vector function at the point 0 ?
b) In which point the value of the vector function is about (3,-3.8)?
c) Which of the following is f(x)?
4cos 2x 3cos 5x 3sin 5x 		don't know
d) Which of the following is g(x)?
4sin 2x 3cos 5x 4cos 2x 		don't know

Vector functions: movement in the plane
Problem 20
Beside we observe time (t) dependent movement of a certain particle in the plane.
This page requires a Java capable browser. Try the meaning of the buttons, reset if needed.
In the right lower corner there is a "ruler", with which you can measure distances in the picture.
Tell the following things about the system:
a) When is the particle nearest the origin?
b) How close to the origin it is then?
c) When is the particle most far away from the origin?
d) How far is it then?
e) The movement is periodic, what is the length of the basic (shortest) period?

IV Student Feedback

Problem 21.

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Updated 17.3.2016