3.4 Rational function  (Page 6/6)

 Page 6 / 6

Range of rational function

The set of real values of rational polynomial for real values of x need not be the range of the function. It is because rational function is not defined for zeroes of polynomial in denominator. In previous section, we evaluated values of function for real x. But, domain may not be the real number set, but subset of R, which excludes certain values of x. We need to exclude values of “y”, which corresponds to values of x for which denominator becomes zero. This statement, however, is slightly confusing, because function is not defined for those values of x in the first place. How would we determine values of y corresponding to values of x for which function reduces to indeterminate form involving division by zero. We actually determine limiting values of function at these points and exclude those values of y from the real set of y, which is determined assuming x belonging to R.

There are certain cases in which denominator of the rational function can not become zero. Consider rational functions :

$f\left(x\right)=\frac{2{x}^{2}-x+1}{{x}^{2}+1}$ $g\left(x\right)=\frac{x+1}{2{x}^{2}-x+1}$ $h\left(x\right)=\frac{2{x}^{2}-x+1}{|x|+1}$

The denominators of all these functions can not be zero. Under this condition, domain of the function is real number set R.

Problem : Find the range of function :

$f\left(x\right)=\frac{x}{1+{x}^{2}}$

Solution : The denominator of the given rational function can not be zero. Hence, domain of function is real number set R. There is no exclusion point. Rearranging to form a quadratic equation in x, we have :

$⇒y+y{x}^{2}=x$

$⇒y{x}^{2}-x+y=0$

We should analyze for coefficient of “ ${x}^{2}$ ” in the quadratic equation. For quadratic equation, coefficient of “ ${x}^{2}$ ” can not be zero i.e. y ≠ 0. For real x, y ≠ 0 and D≥0 :

$D={\left(-1\right)}^{2}-4XyXy=1-4{y}^{2}\ge 0$ $⇒{y}^{2}\ge \frac{1}{4}$ $⇒y\in \left[-\frac{1}{2},\frac{1}{2}\right]$

What if y=0? Putting this value in the quadratic equation, we have :

$⇒0-x+0=0$

$⇒x=0$

This is included in the domain. Hence, y=0 is included in the range. The range of the rational function, therefore, remains unaffected :

$⇒y\in \left[-\frac{1}{2},\frac{1}{2}\right]$

Problem : Find the range of the function :

$y=f\left(x\right)=\frac{{x}^{2}-5x+4}{{x}^{2}-3x+2}$

Solution : We see that discrimanants of numerator and denominator polynomials are positive. On factorizing,

$⇒y=\frac{{x}^{2}-5x+4}{{x}^{2}-3x+2}=\frac{\left(x-1\right)\left(x-4\right)}{\left(x-1\right)\left(x-2\right)}$

Clearly, rational function is not defined for x=1 and x=2. Domain of the function is R- {1,2). For the sake of determining range, the limiting values of function for these values of x are obtained by canceling (x-1) from numerator and denominator :

$⇒y=\frac{\left(x-4\right)}{\left(x-2\right)}$

For x=1, y = 3. For x=2, however, the function value is indeterminate. In totality, we need to exclude y=3 from the interval of real values of y. Now, in order to determine real values of y, we rearrange the given function to form a quadratic equation in x :

$⇒y{x}^{2}-3yx+2y={x}^{2}-5x+4$ $⇒\left(y-1\right){x}^{2}+\left(5-3y\right)x+2y-4=0$

We should analyze for coefficient of “ ${x}^{2}$ ” in the quadratic equation. For quadratic equation, coefficient of “ ${x}^{2}$ ” can not be zero i.e. y-1 ≠ 0. For real x, y-1 ≠ 0 and D≥0.

For y-1 = 0, y = 1. Putting this value in the quadratic equation,

$⇒0+\left(5-3\right)x+2-4=0$ $⇒x=1$

We see that x=1 is not part of domain. This is actually the value which reduces denominator to zero. Hence, we should exclude y = 1 from the real values of y. Now for D≥0,

$D={\left(5-3y\right)}^{2}-4\left(y-1\right)\left(2y-4\right)\ge 0$ $⇒25+9{y}^{2}-30y-4\left\{2{y}^{2}-6y+4\right\}\ge 0$ $⇒25+9{y}^{2}-30y-4\left\{2{y}^{2}-6y+4\right\}\ge 0$

The coefficient of ${y}^{2}$ is positive. The discriminant is 0. Clearly, following sign rule, f(x) ≥0 for all real values of y. Hence, real values of y are real number set R. However, we need to exclude y = {1,3) as discussed above. Therefore, range of given function is R-{1,3}.

Alternative

Once, exception points are noted, we can evaluate “y” from the reduced form :

$⇒y=\frac{\left(x-4\right)}{\left(x-2\right)}$

Solving,

$⇒x=\frac{2y-4}{\left(y-1\right)}$

Clearly, y#1. But we have seen that y#3 as well. Hence, range of rational function is R-{1,3}.

Graph of rational function

We know that rational function is a composition of two functions in the following form,

$f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$

where q(x) ≠ 0. If q(x) = 0, then the ratio has the form “ $x/0$ ”, which is not defined.

For plotting, let us consider a simple rational function given by, $f\left(x\right)=1/x$ . This function is known as reciprocal function. It is not defined for x = 0. In order to plot the function, we calculate few initial values as :

$For\phantom{\rule{1em}{0ex}}x=-1,\phantom{\rule{1em}{0ex}}y=-1$

$For\phantom{\rule{1em}{0ex}}x=-2,\phantom{\rule{1em}{0ex}}y=-0.5$

$For\phantom{\rule{1em}{0ex}}x=-3,\phantom{\rule{1em}{0ex}}y=-0.33$

$For\phantom{\rule{1em}{0ex}}x=0,\phantom{\rule{1em}{0ex}}\text{y is not defined}$

$For\phantom{\rule{1em}{0ex}}x=1,\phantom{\rule{1em}{0ex}}y=1$

$For\phantom{\rule{1em}{0ex}}x=2,\phantom{\rule{1em}{0ex}}y=0.5$

$For\phantom{\rule{1em}{0ex}}x=3,\phantom{\rule{1em}{0ex}}y=0.33$

The graph of the function is shown here :

This plot is not defined at x = 0. The domain of the given function, therefore, is real numbers, “R” except zero. Also,

$⇒x=\frac{1}{y}$

This means that function value can not be zero. Hence, range of the function is also real numbers, “R” except zero.

$Domain=R-\left\{0\right\}$

$Range=R-\left\{0\right\}$

Problem : Draw the graph of rational function given by :

$f\left(x\right)=\frac{{x}^{2}-1}{x-1}$

Discuss the nature of graph and also determine domain and range of the given function.

Solution : The form of the given function is that of rational function. We observe that the function is not defined for "x = 1" as function has the form " $x/0$ ", which is undefined. The domain of the given function, therefore, is “R” except “1”. It should be noted that while interpreting domain or range we should not cancel out common terms in the numerator and denominator.

For other values of “x”, the value of the function is given by the reduced expression :

$f\left(x\right)=\frac{{x}^{2}-1}{x-1}=x+1$

Clearly, if the given function were valid for x =1, then y = x+1 = 1 + 1 = 2. Thus, function f(x) can take any real value except “2”. Hence, range of the function is "R" except "2". The domain and range of the given function are :

$Domain=R-\left\{1\right\}$

$Range=R-\left\{2\right\}$

In order to plot the function, we calculate few initial values as :

$For\phantom{\rule{1em}{0ex}}x=-3,\phantom{\rule{1em}{0ex}}y=-2$

$For\phantom{\rule{1em}{0ex}}x=-2,\phantom{\rule{1em}{0ex}}y=-1$

$For\phantom{\rule{1em}{0ex}}x=-1,\phantom{\rule{1em}{0ex}}y=0$

$For\phantom{\rule{1em}{0ex}}x=0,\phantom{\rule{1em}{0ex}}y=1$

$For\phantom{\rule{1em}{0ex}}x=1,\phantom{\rule{1em}{0ex}}\text{y is not defined}$

$For\phantom{\rule{1em}{0ex}}x=2,\phantom{\rule{1em}{0ex}}y=3$

$For\phantom{\rule{1em}{0ex}}x=3,\phantom{\rule{1em}{0ex}}y=4$

The graph of the function is shown here :

The plot is not defined at x = 1. There is a break at x = 1.

Nature of graph

Here, we consider graphs of rational functions of type :

$y=\frac{1}{x},\phantom{\rule{1em}{0ex}}\frac{1}{{x}^{3}},\phantom{\rule{1em}{0ex}}\frac{1}{{x}^{5}},.............$

The nature of graph of these rational function of type $y=\frac{1}{{x}^{n}}$ , where n is an odd integer such that n≥ 1, is similar to graph of y=1/x as shown in the figure. The graph is that of rectangular hyperbola.

We need to emphasize that the graph generalizes the nature and is helpful to estimate domain and range of functions. We need to graph individual function if required. The nature of graph of function type $y=\frac{1}{{x}^{n}}$ , where n is an even integer such that n≥2 is shown in the figure below :

$y=\frac{1}{{x}^{2}},\phantom{\rule{1em}{0ex}}\frac{1}{{x}^{4}},\phantom{\rule{1em}{0ex}}\frac{1}{{x}^{4}},.............$

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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  By By Edward Biton By   By By Rhodes  