5.8 One-one and many-one functions (exercise)

 Page 1 / 1

A function is one – one function, if every pre-image(x) in domain is related to a distinct image (y) in the co-domain. Otherwise, function is many-one function.

Note : This module did not follow the treatment on the subject as we needed to know different types of real functions in the first place. Besides, there are additional methods to determine function types. In particular, the concept of monotonous functions (increasing or decreasing) can be used to determine whether a function is one-one or not.

Working rules

• Put $f\left({x}_{1}\right)=f\left({x}_{2}\right)$ . Solve equation. If it yields ${x}_{1}={x}_{2}$ , then function is one-one; otherwise not.
• Alternatively, draw plot of the given function. Draw a line parallel to x-axis such that it intersects as many points on the plot as possible. If it intersects the graph only at one point, then the function is one-one.
• Alternatively, put f(x) = 0. Solve f(x) = f(0) for “x” and see whether “x” is single valued for being one-one function.
• Alternatively, a function is an one-one function, if f(x) is a continuous function and is either increasing or decreasing function in the given domain.

Problem 1: A function $f:R\to R$ is given by :

$f\left(x\right)={x}^{3}$

Is the function one-one.

Solution :

Statement of the problem : Draw a line parallel to x-axis to intersect the plot of the function as many times as possible.

We find that all lines drawn parallel to x-axis intersect the plot only once. Hence, the function is one-one.

Problem 2: A function $f:R\to R$ is given by :

$f\left(x\right)={x}^{2}$

Is the function one-one.

Solution :

Statement of the problem : We can solve $f\left({x}_{1}\right)=f\left({x}_{2}\right)$ and see whether ${x}_{1}={x}_{2}$ to decide the function type.

$⇒{x}_{1}^{2}={x}_{2}^{2}$

${x}_{1}=±{x}_{2}$

We see that " ${x}_{1}$ " is not exclusively equal to " ${x}_{2}$ ". Hence, given function is not one-one function, but many – one function. This conclusion is further emphasized by the intersection of a line parallel to x-axis, which intersects function plot at two points.

Problem 3: A function $f:R\to R$ is given by :

$f\left(x\right)=x|x|$

Is the function one-one.

Solution :

Statement of the problem : Draw the plot of the function and see intersection of a line parallel to x-axis.

We observe from its plot that there is no line parallel to x-axis, which intersects the functions more than once. Hence, function is one-one.

Problem 4: Determine whether greatest integer function is one-one function.

Solution : Statement of the problem : Draw the plot of the function and see intersection of a line parallel to x-axis.

We have drawn one such line at y = 1. We see that this function value is valid for an interval of “x” given by 1≤x<2. Hence, greatest integer function is not one-one, but many –one function.

Problem 5: A function $f:R\to R$ is given by :

$f\left(x\right)=\frac{{x}^{2}+4x+30}{{x}^{2}-8x+18}$

Is the function one-one.

Solution :

Statement of the problem : The given function is a rational function. We have to determine function type.

We evaluate function for x =0. If f(x)=f(0) equation yields multiple values of “x”, then function in not one-one. Here,

$⇒f\left(0\right)=\frac{{x}^{2}+4x+30}{{x}^{2}-8x+18}=\frac{{0}^{2}+4X0+30}{{0}^{2}-8X0+18}=\frac{30}{18}=\frac{5}{3}$

Now,

$⇒f\left(x\right)=f\left(0\right)$

$⇒f\left(x\right)=\frac{{x}^{2}+4x+30}{{x}^{2}-8x+18}=\frac{5}{3}$

$⇒3{x}^{2}+12x+90=5{x}^{2}-40x+90\phantom{\rule{1em}{0ex}}⇒2{x}^{2}-52x=0$

$⇒x=0,26$

We see that f(0) = f(26). It means pre-images are not related to distinct images. Thus, we conclude that function is not one-one, but many-one.

Problem 6 : A continuous function $f:R\to R$ is given by :

$f\left(x\right)=\frac{{x}^{2}+4x+30}{{x}^{2}-8x+18}$

Determine increasing or decreasing nature of the function and check whether function is an injection?

Solution :

Statement of the problem : The rational function is a continuous function. Hence, we can determine its increasing or decreasing nature in its domain by examining derivative of the function.

$⇒f\prime \left(x\right)=\frac{\left({x}^{2}-8x+18\right)\left(2x+4\right)-\left({x}^{2}+4x+30\right)\left(2x-8\right)}{{\left({x}^{2}-8x+18\right)}^{2}}$

$⇒f\prime \left(x\right)=\frac{-12\left({x}^{2}+2x-26\right)}{{\left({x}^{2}-8x+18\right)}^{2}}$

The denominator is a square of a quadratic expression, which evaluates to a positive number. On the other hand, the discreminant of the quadratic equation in the numerator is :

$⇒D={\left(2\right)}^{2}-4X1X\left(-26\right)=4+104=108$

It means that derivative has different signs in the domain interval. Therefore, the function is a combination of increasing and decreasing nature in different intervals composing domain. Thus, function is not monotonic in the domain interval. Hence, we conclude that function is not an injection.

Problem 7: A function $f:R\to R$ is given by :

$f\left(x\right)=\mathrm{cos}\left(3x+2\right)$

Is the function one-one.

Solution :

Statement of the problem : We solve $f\left({x}_{1}\right)=f\left({x}_{2}\right)$ and see whether ${x}_{1}={x}_{2}$ to decide the function type.

$\mathrm{cos}\left(3{x}_{1}+2\right)=\mathrm{cos}\left(3{x}_{2}+2\right)$

The basic solution is :

$⇒3{x}_{1}+2=±\left(3{x}_{2}+2\right)$

General solution is obtained by adding integral multiples of the period of function, which is “2π” for cosine function :

$⇒3{x}_{1}+2=2n\pi ±\left(3{x}_{2}+2\right);\phantom{\rule{1em}{0ex}}n\in Z$

For n = 1,

$⇒f\left(3{x}_{1}+2\right)=f\left\{2\pi ±\left(3{x}_{2}+2\right)\right\}$

Thus, multiple pre-images are related to same image. Hence, given function is not one-one.

We can easily interpret from the plots of different trigonometric functions that they are not one-one functions. However, they are one-one in the subset of their domain. Such is the case with other functions as well. They are generally not one-one, but may reduce to one-one in certain interval(s).

Problem 8 : A function $f:R\to R$ is given by :

$f\left(x\right)=\frac{a{x}^{2}+6x-8}{a+6x-8{x}^{2}}$

Is the function one-one for a = 3?

Solution :

Statement of the problem : The given function is a rational function. Each of numerator and denominator functions is quadratic equation. In this case, solving f(a) = 0 for “x” reveals the nature of function for a =3.

$⇒f\left(x\right)=\frac{3{x}^{2}+6x-8}{3+6x-8{x}^{2}}=0$

$⇒3{x}^{2}+6x-8=0$

$⇒x=\frac{-6±\sqrt{\left(36-4X3X-8\right)}}{6}=-1±\frac{\sqrt{33}}{3}$

As “x” is not unique, the given function is not one-one function. We should emphasize here that solution of function when equated to zero is not a full proof method. In this particular case, it turns out that function value becomes zero for two values of “x”. In general, we should resort to techniques outlined in the beginning of the module to determine function type.

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x