# 4.2 Transformation of graphs by modulus function  (Page 4/4)

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$y=f\left(x\right)\phantom{\rule{1em}{0ex}}⇒\phantom{\rule{1em}{0ex}}x=|f\left(y\right)|$

The invertible function x= f(y) has its inverse function given by y=f⁻¹(x). Alternatively, if a function is defined as y=f⁻¹(x), then variables x and y are related to each other such that x=f(y). We conclude that graph of y=f⁻¹(x) is same as graph of x=f(y) with the same orientation of x and y axes. It is important to underline here that we transform (change) graph of inverse of given function i.e. y=f⁻¹(x) to get the transformation of graph of x=f(y). Further x and y coordinates on the graph correspond to x and y values.

We interpret assignment of |f(y)| to x in the given graph in accordance with the definition of modulus function. Consider x=|f(y)|. But, modulus can not be equated to negative value. Hence, x can not be negative. It means we need to discard left half of the graph of inverse function y=f⁻¹(x). On the other hand, modulus of negative or positive value is always positive. Hence, positive value of x=a correspond to two values of function in dependent variable, a=±f(y). Corresponding to these two function values in y, we have two values of y i.e. f⁻¹(a) and f⁻¹(-a). In order to plot two values, we need to take mirror image of the left half of the graph of y=f⁻¹(x) across y-axis. This is image in y-axis.

From the point of construction of the graph of x=|f(y)|, we need to modify the graph of y=f⁻¹(x) i.e. x=f(y) as :

1 : take mirror image of left half of the graph in y-axis

2 : remove left half of the graph

This completes the construction for x=|f(y)|.

Problem : Draw graph of $x=|\mathrm{cosec}y|;\phantom{\rule{1em}{0ex}}x\in \left\{-\pi /2,\pi /2\right\}$ .

Solution : The inverse of base function is cosec⁻¹x. We first draw the graph of inverse function. Then, we take mirror image of left half of the graph in y-axis and remove left half of the graph to complete the construction of graph of $x=|\mathrm{cosec}y|$ .

## Examples

Problem : Find domain of the function given by :

$f\left(x\right)=\frac{1}{\sqrt{|\mathrm{sin}x|+\mathrm{sin}x}}$

Solution : The square root gives the condition :

$⇒|\mathrm{sin}x|+\mathrm{sin}x\ge 0$

But denominator can not be zero. Hence,

$⇒|\mathrm{sin}x|+\mathrm{sin}x>0$

$⇒|\mathrm{sin}x|>-\mathrm{sin}x$

We shall make use of graphing technique to evaluate the interval of x. Since both functions are periodic. It would be indicative of the domain if we confine our consideration to 1 period of sine function (0, 2π) and then extend the result subsequently to other periodic intervals.

We first draw sine function. To draw |sinx|, we take image of lower half in x-axis and remove the lower half. To draw “–sinx”, we take image of y=sinx in x-axis.

From the graph, we see that |sinx| is greater than “-sinx” in (0,π). Note that end points are not included. The domain is written with general notation as :

$x\in \left(2n\pi ,\left(2n+1\right)\pi \right)$

Problem : Determine graphically the points where graphs of $|y|={\mathrm{log}}_{e}|x|$ and ${\left(x-1\right)}^{2}+{y}^{2}-4=0$ intersect each other.

Solution : The function $|y|={\mathrm{log}}_{e}|x|$ is obtained by transforming $y=\mathrm{log}{}_{e}x$ . To draw $y={\mathrm{log}}_{e}|x|$ , we need to remove left half (but here there is no left half) and take image of right half in y-axis. To draw $|y|=\mathrm{log}{}_{e}|x|$ , we transform the graph of $y=\mathrm{log}{}_{e}|x|$ . For this, we remove the lower half and take image of upper half in x-axis.

On the other hand, ${\left(x-1\right)}^{2}+{y}^{2}-4=0$ is a circle with center at 1,0 having radius of 2 units. Finally, superposing two graphs, we determine the intersection points.

Clearly, there are three intersection points as shown by solid circles.

## Exercises

Draw the graph of function given by :

$f\left(x\right)=\frac{1}{\left[x\right]-1}$

Hints : Draw 1/x, which is a hyperbola with center at (0,0). Then draw 1/x-1. It is a hyperbola shifted right by 1 unit. Its center is (1,0). Remove left half and take the image of right half in y-axis.

2. Draw the graph of function given by :

$f\left(x\right)=||\frac{1}{x}|-1|$

Hints : Draw 1/x, which is a hyperbola with center at (0,0). Then draw |1/x|. Take image of lower half in x-axis. Remove lower half. To draw |1/x|-1, shift down the graph of |1/x| by 1 unit. To draw ||1/x|-1|, Take image of lower half of the graph of |1/x|-1 in x-axis. Remove lower half.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
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Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x

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