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Graph of f(x)=1/x with its vertical asymptote at x=0.

Vertical asymptote

A vertical asymptote    of a graph is a vertical line x = a where the graph tends toward positive or negative infinity as the inputs approach a . We write

As  x a , f ( x ) ,   or as  x a , f ( x ) .

End behavior of f ( x ) = 1 x

As the values of x approach infinity, the function values approach 0. As the values of x approach negative infinity, the function values approach 0. See [link] . Symbolically, using arrow notation

As  x , f ( x ) 0 , and as  x , f ( x ) 0.

Graph of f(x)=1/x which highlights the segments of the turning points to denote their end behavior.

Based on this overall behavior and the graph, we can see that the function approaches 0 but never actually reaches 0; it seems to level off as the inputs become large. This behavior creates a horizontal asymptote , a horizontal line that the graph approaches as the input increases or decreases without bound. In this case, the graph is approaching the horizontal line y = 0. See [link] .

Graph of f(x)=1/x with its vertical asymptote at x=0 and its horizontal asymptote at y=0.

Horizontal asymptote

A horizontal asymptote    of a graph is a horizontal line y = b where the graph approaches the line as the inputs increase or decrease without bound. We write

As  x  or  x ,   f ( x ) b .

Using arrow notation

Use arrow notation to describe the end behavior and local behavior of the function graphed in [link] .

Graph of f(x)=1/(x-2)+4 with its vertical asymptote at x=2 and its horizontal asymptote at y=4.

Notice that the graph is showing a vertical asymptote at x = 2 , which tells us that the function is undefined at x = 2.

As  x 2 , f ( x ) ,  and as  x 2 + ,   f ( x ) .

And as the inputs decrease without bound, the graph appears to be leveling off at output values of 4, indicating a horizontal asymptote at y = 4. As the inputs increase without bound, the graph levels off at 4.

As  x ,   f ( x ) 4  and as  x ,   f ( x ) 4.
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Use arrow notation to describe the end behavior and local behavior for the reciprocal squared function.

End behavior: as x ± ,   f ( x ) 0 ; Local behavior: as x 0 ,   f ( x ) (there are no x - or y -intercepts)

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Using transformations to graph a rational function

Sketch a graph of the reciprocal function shifted two units to the left and up three units. Identify the horizontal and vertical asymptotes of the graph, if any.

Shifting the graph left 2 and up 3 would result in the function

f ( x ) = 1 x + 2 + 3

or equivalently, by giving the terms a common denominator,

f ( x ) = 3 x + 7 x + 2

The graph of the shifted function is displayed in [link] .

Graph of f(x)=1/(x+2)+3 with its vertical asymptote at x=-2 and its horizontal asymptote at y=3.

Notice that this function is undefined at x = 2 , and the graph also is showing a vertical asymptote at x = 2.

As  x 2 ,   f ( x ) , and as   x 2 + ,   f ( x ) .

As the inputs increase and decrease without bound, the graph appears to be leveling off at output values of 3, indicating a horizontal asymptote at y = 3.

As  x ± ,   f ( x ) 3.
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Sketch the graph, and find the horizontal and vertical asymptotes of the reciprocal squared function that has been shifted right 3 units and down 4 units.

Graph of f(x)=1/(x-3)^2-4 with its vertical asymptote at x=3 and its horizontal asymptote at y=-4.

The function and the asymptotes are shifted 3 units right and 4 units down. As x 3 , f ( x ) , and as x ± , f ( x ) 4.

The function is f ( x ) = 1 ( x 3 ) 2 4.

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Solving applied problems involving rational functions

In [link] , we shifted a toolkit function in a way that resulted in the function f ( x ) = 3 x + 7 x + 2 . This is an example of a rational function. A rational function is a function that can be written as the quotient of two polynomial functions. Many real-world problems require us to find the ratio of two polynomial functions. Problems involving rates and concentrations often involve rational functions.

Questions & Answers

linear speed of an object
Melissa Reply
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
Melissa
how to find domain
Mohamed Reply
like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
Dan
define the term of domain
Moha
if a>0 then the graph is concave
Angel Reply
if a<0 then the graph is concave blank
Angel
what's a domain
Kamogelo Reply
The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.
Spiro
how fast can i understand functions without much difficulty
Joe Reply
what is inequalities
Nathaniel
functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2
Dan
what is set?
Kelvin Reply
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
Divya Reply
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49
Dan
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2
Dan
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
Practice Key Terms 5

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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