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f x = sec x

Again, the function is not defined for all real number “x”. Let us recall that :

sec x = 1 cos x

This is a rational polynomial form, which is defined for cos x 0 . Now, cos(x) evaluates to zero for values of “x”, which appears at a certain interval given by the condition,

cos x = 0 ; x = 2 n + 1 π 2 , where n Z

The function cos(x) is zero for x = ± π 2, ± 3 π 2, ± 5 π 2, etc. In other words, the cosine function is zero for all odd multiples of “ π / 2 ”. It means that secant function is not defined for odd multiples of “ π / 2 ”. These values of “x”, for which cosine is zero, need to be excluded from real number set “R”.

The values of secant function are bounded by certain intervals. We have seen that values of cosine function is between “-1” and “1”, including end points. Just like the case of cosecant function, the range of secant function is :

| sec x | 1

or

( - , - 1 ] ( 1, ]

Hence, domain and range of secant function are :

Domain = R { x : x = 2 n + 1 π / 2 , n Z }

Range = ( - , - 1 ] ( 1, ]

The plot of secant(x) .vs. x is shown here.

Secant function

Graph of secant function

The period of secx is 2π. Important to note here is that function is not defined even within a periodic segment. Since sec(-x) = secx, we conclude that secant function is even function in each of periodic segment. Multiplication of secant function by a constant A changes the range plot lies on or beyond -A or A. The range is modified as :

Range = ( - , - A ] ( A, ]

Multiplying argument x like sec(kx), however, changes the points where function is not defined. It is now given by :

x = 2 n + 1 π / 2k , n Z

Therefore, domain is now modified as :

Domain = R { x : x = 2 n + 1 π / 2k , n Z }

Cotangent function

For a real number “x”, there is a cotangent function defined as :

f x = cot x

The function is not defined for all real number “x”. Let us recall that :

cot x = cos x sin x

This is a rational polynomial form, which is defined for sin x 0 . Now, sin(x) evaluates to values of “x”, which appears at a certain interval given by the condition,

sin x = 0 ; x = n π , where n Z

This means that sin(x) is zero for x = 0, ± π , ± 2 π , ± 3 π , etc. In other words, the sine function is zero for all integral multiples of “ π ”. It means that cotangent function is not defined for integral multiples of “ π ”. Values of “x”, for which sine is zero, need to be excluded from real number set “R”. On the other hand, the values of cotangent function are extended along the real number line on either side of zero. The range of the function, therefore, is “R”. Hence, domain and range of cotangent function are :

Domain = R { x : x = n π , n Z }

Range = R

The plot of cot(x) .vs. x is shown here.

Cotangent function

Graph of cotangent function

The period of cotx is π. Since cot(-x) = -cotx, we conclude that cotangennt function is odd function in each of periodic segment. Multiplication of cotangennt function by a constant A does not change the range plot extends either side of x-axis. Multiplying argument x like cot(kx), however, changes the points where function is not defined. It is now given by :

x = n π k , n Z

Therefore, domain is now modified as :

Domain = R { x : x = n π k , n Z }

Examples

Problem : Find range of the function :

f x = 1 2 sin 2 x

Solution : The denominator of given function is non-negative as value of sin2x can not exceed 1. We know range of sin2x. We shall build up expression from this basic trigonometric function to determine range of the given function. Here,

- 1 sin 2 x 1

Multiplying with -1 to change sign of sine function, we have (note the change in inequality sign) :

1 - sin 2 x - 1 2 + 1 2 - sin 2 x 2 - 1 3 2 - sin 2 x 1

We need to take reciprocal of each term in the equality to obtain required function form (note the change in inequality sign),

1 3 1 2 sin 2 x 1 1 3 f x 1 Range = 1 3, 1

This is a unique method to determine range by building up a function from a basic function along with change in the interval of values. We need to be careful that such building up of function does not introduce condition in which function becomes indeterminate. Further, we can find range conventionally by solving function for x in terms of y. We have not considered this method here as solution for x involves inverse trigonometric function. We shall, however, revisit this problem subsequent to the study of inverse trigonometric function.

Problem : Find range of function :

f x = 2 sin π 2 4 x 2

Solution : Before we attempt to find range, we need to find domain of the function so that we can determine interval of function values. We know that expression within square root is non-negative. Also, expression is a quadratic function. Analyzing this quadratic function, domain of quadratic function is found as [-π/2, π/2]. Coefficient of squared term is negative. Hence, its graph opens down and maximum value of quadratic function is :

y max = D 4 a = 0 - 4 X - 1 X π 2 4 X - 1 = π 2

Since expression is non-negative within square root, minimum value of function is 0 (see figure). Now, sine function is an increasing function in [0, π/2] as is evident from its graph. Thus sine function assumes values in the interval [sin0, sin π/2]i.e. [0,1]. Sine function, however, has a coefficient of 2 . As such, range of given function is [0,2].

Exercise

Problem : Find the domain of the function given by :

f x = cos sin x

Solution : The argument (input) to cosine function is sine function. The expression within square root is non-negative. It means that :

cos sin x 0

We know that cosine function is positive in first and fourth quadrants. It means that the argument of the cosine function should be between -π/2 and π/2. Therefore, we need to see whether the value of “sinx” falls within this interval or not? The value of sine function, on the other hand, lies in the interval [-1,1]. This is indeed (as shown in the figure below) within the required interval for cosine function to be non-negative as 1<π/2 and -1>-π/2.

Domain of cosine function

The range of sine function falls within domain of cosine function

Now, we know that sine function is real for all real values of “x”. Hence, domain of the given function is :

Domain = - ,

Problem : Find the domain of the function given by :

f x = cos 2 π [ x 1 ]

Solution : The cosine function is valid for all real values of its argument. The argument, however, is in rational form, requiring that denominator is not zero. Hence,

[ x - 1 ] 0

We can easily evaluate this inequality knowing the fact that greatest integer function is zero in the interval 0≤x<1. This is also substantiated by the graph of greatest integer function as shown here. Now, applying to the greatest integer function of the denominator, the interval in which greatest integer function is equal to zero is :

Greatest integer function

The greatest integer function [x] returns zero in the interval 0≤x<1.

0 x - 1 < 1 1 x < 2

Hence, domain of the given function is :

Domain = R [ 1,2 )

Check the validity of the composition and find domain of the function given by :

f x = sin [ log e { 9 x 2 1 x } ]

We know that range of logarithmic function is “R”. Here, logarithmic function itself is the argument of sine function. This means that argument of sine function is "R". This meets the requirement of a sine function, whose domain is "R". Hence, composition of function as given in the question is a valid composition.

In order to find, domain of the function, we need to find values of “x” for which argument of logarithmic function is a positive real number. In the nutshell, we need to evaluate following inequality :

9 x 2 1 x > 0

Here, we see that numerator is a square root of a polynomial and is, therefore, positive. Now, evaluating of the polynomial in the numerator for positive real number, we have :

9 x 2 > 0

x 2 9 < 0

x + 3 x 3 < 0

- 3 < x < 3

Since numerator is positive, the denominator needs to be positive so that total rational polynomial is positive. Evaluating inequality relation for the polynomial in the denominator for positive real number :

1 x > 0

x < 1

Now, we know that the domain of the quotient is D 1 D 2 . Hence,

Domain = ( - 3 < x < 3 ) ( x < 1 )

Domain of "f" = - 3 < x < 1

Find range of function :

f x = cos x 2 π 2 9

Range is [-1,1]

Find range of function :

f x = tan π 2 9 x 2

Range is [0,√3]

Questions & Answers

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
What is power set
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Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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