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Candy Su

on 20 May 2014

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Brain at work

Domain is the most important thing about a function. The importance of domain becomes even more obvious in rational functions as the restrictions of the function will lead to the formation of vertical asymptote or hole.
Introducing Properties of Rational Functions:

the form of
, where
f(x) & g(x) are polynomial
functions and g(x)≠0
The zeros of a rational function h(x)= f(x)/g(x) are the zeros of f(x) if h(x) is expressed in simplified form.
ex: f(x)= x(x-2)/(x+2)(x-1) will have zeros of 0 and 2

From the graph of f(x)=x(x-2)/(x+2)(x-1), it demonstrates that zeros occur when f(x) equals 0, which also means that zeros will occur if the numerator of a rational function equals 0.
Introducing Properties of Rational Functions:
the y-intercept of a rational function h(x) is the value of the simplified form of the function when x=0.
ex: h(x)= (x+2)(x-5)(x-3)/(x+3)(x-2)(x-6), y-intercept will be: 5/6

From the graph of h(x)= (x+2)(x-5)(x-3)/(x+3)(x-2)(x-6), it demonstrates that y-intercept will occur when x equals 0
The domain of a rational function consists of all the real numbers except for the zeros of the denominator.

{XER|X≠ 1 and X≠ -2}
Domain of f(x):
Domain of h(x):
{XER|X≠-3 and x≠2 and x≠6}
Restrictions lead to asymptotes/holes:
What is an Asymptote?

An asymptote is a line or curve that approaches a given curve arbitrarily
Like this:

If there is
would there also be finite(removable)?
Yes, there can also be a removable discontinuity instead of a infinite discontinuity, removable discontinuity is also known as holes.

Removable discontinuity usually occurs when the function can be simplified by dividing the denominator and numerator by a common factor that includes a variable.

Connection between restrictions and asymptotes/holes:
Therefore, at rational function's restriction values, a rational function can either have a removable/ infinite discontinuity or both.

a hole
a vertical asymptote
To sum up...
The connections between a vertical asymptote and a hole are that vertical asymptote and hole both came from
which means they are also
yet the major difference between a vertical asymptote and a hole is that
vertical asymptote is an infinite discontinuity
hole is a removable discontinuity.


between horizontal and vertical asymptote
Imaginary line
A function can go across its horizontal asymptote while it cannot go across its vertical asymptote
Horizontal asymptote mimics the behavior of the function while vertical asymptote came from the restrictions
Obtain horizontal asymptote through division; vertical asymptote through restrictions


between horizontal and vertical asymptote
Vertical asymptote is always straight, yet horizontal asymptote can be slanted or even in a parabolic shape
A function can have more than 1 vertical asymptote and also more than 1 horizontal asymptote
An asymptote is a line or curve that approaches a given curve arbitrarily close
H.A answers the question "what does f(x) do x gets infinitely large in either direction?"
Serve as an End behavior asymptote
Process of obtaining a H.A
We use division to obtain the H.A (E.B.A) of a rational function:

Through division, we can sum up that:
Degree of numerator < Degree of denominator
Horizontal occurs at f(x)=0
ex: f(x)= x/x^2

Degree of numerator = degree of denominator
H.A will be the ratio of the leading coefficient
ex: f(x)= 2(x-1)/x; H.A occurs when f(x)=2

Degree of numerator is exactly 1 degree greater than the denominator: Oblique(slant) asymptote
ex: f(x)= x^2/(x+1)

H.A occurs when f(x)= x-1
Degree of numerator is exactly 2 degrees greater than denominator: Parabolic asymptote
ex: x^3/(x+1)

H.A occurs when f(x)= x^2+1
Point of Intersections
Different behavior near asymptotes
f(x)= 1/x
Determining V.A and the behavior near V.A

Therefore, As x--> 0-, f(x) --> negative infinity
As x--> 0+, f(x) --> infinity

Determining E.B.A and describe its behavior
As x --> infinity, f(x) --> 0 (approaching from above)
As x --> negative infinity, f(x) --> 0 (from below)
Extensive Question:

Let’s look at this question with the function of y=x and the reciprocal function of y = 1/x.

1. When x equals 0 as a denominator, the value of y is undefined, therefore, the reciprocal function does not goes through the point (0,0) like the function ‘y = x’ does. Therefore, the domain of the reciprocal transformation will be: {XER|X ≠ 0} and the range will be: {YER| Y ≠ 0}. The domain and the range of the reciprocal transformation demonstrates that as |x| get closer to the vertical asymptote (which is 0 in this case), the absolute value of y becomes greater. However, as |x| move further away from the vertical asymptote, the absolute value of y becomes smaller (also never touches the asymptote). Therefore, the reciprocal transformation of y= 1/x will never go across its asymptote to form a straight line like the function y = x does.
2. There can only be two points where the values of ‘x’ and ‘y’ are both whole numbers in a reciprocal transformation, which is when the absolute value of the denominator equals the value of the numerator (ex: (1,1) (-1,-1) in the case of y=1/x). As a result, curved lines must occur in a reciprocal transformation, because if the value of ‘x’ is a whole number, the value of ‘y’ will never be a whole number (unless the value of ‘x’ equals the value of the numerator); yet, when the value of ‘y’ is a whole number, the value of ‘x’ cannot be a whole number, meaning that a straight line will not occur.
Moreover, due to the fact that a straight line have a constant slope and a reciprocal function may have an increasing slope or decreasing slope; thus, reciprocal transformation give curved lines when the original function has a straight line.

If rational function can cross its E.B.A, what will be the point of intersection?
This question will lead to solving EQUALITIES...
for example, solve:
, point of intersection?

Solving inequalities is similar as determining behavior near V.A, since both of them requires an interval table. For example,

After learning rational functions, as a student, we may wonder, what is the connection between rational functions and real-life? When will we ever use rational functions?
Well, here are some real-life examples:
Electricity that we use everyday:

As a biologist:

In medical field:

The topic rational function becomes more and more significant in our society as doctors are required to calculate the concentration of a drug injected in a patient in order to perform the right medical procedures.
Rational function is also important in the medical field as it portrays how far the drug has affected the patient before starting a treatment, since the outcome can be influenced by the concentration of the drug.
For example, surgeries and operations can only begin after a concentration of a drug reaches a desired level. Therefore, rational function is helpful for doctors to determine after how many hours they should start surgery based on the concentration level they prefer.
By Candy Su
Therefore, Real-life first, mathematics second.
In other words,

Thank you.
Ms. Mendaglio

Background Music:
Brahms - Waltz in A flat Op. 39, No.15
Full transcript