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Here are a few recommended readings before getting started with this lesson.
Identify the asymptotes, domain, and range of each function.
Asymptotes  Domain  Range  

Function I  
Function II  
Function III 
$f(x)=x1 ,x =0$
The graph of the function $f(x)=x1 $ is a hyperbola, which consists of two symmetrical parts called branches. It has two asymptotes, the $x$ and $y$axes. The domain and range are all nonzero real numbers.
The graph of $y=x1 $ can be used to graph other reciprocal functions. This can be done by applying different transformations.
Name  Equation  Characteristics 

Parent Reciprocal Function  $y=x1 $  $Domain:Range:Asymptotes: R−{0}R−{0}xandyaxes $

Inverse Variation Functions  $y=xa $  
General Form of Reciprocal Functions  $y=x−ha +k$  $Domain:Range:Asymptotes: R−{h}R−{k}x=handy=k $

It is important not to confuse the reciprocal of a function with the inverse of a function. For numbers, $a_{1}$ refers to the reciprocal $a1 ,$ while the notation $f_{1}(x)$ is commonly used to refer to the inverse of a function.
Function, $f$  Reciprocal, $f1 $  Inverse, $f_{1}$ 

$f(x)=2x+10$  $f(x)1 =2x+101 $  $f_{1}(x)=21 (x−10)$ 
$f(x)=x_{3}−5$  $f(x)1 =x_{3}−51 $  $f_{1}(x)=3x+5 $ 
Mark wants to identify the asymptotes, domain, and range of the reciprocal function shown as follows.
Mark states the characteristics of the function.Start by identifying any $x$values for which $f(x)$ is undefined.
The branch of the graph to the left of $x=3$ shows that the $y$values approach $4.$ This is true for the other branch as well. Therefore, there is a horizontal asymptote at $y=4.$ Draw the asymptotes on the given coordinate plane.
In the graph, it can be seen that the domain is all real numbers except $3,$ and the range is all real numbers except $4.$ Therefore, Mark's correct statements are III and IV.
For the indicated reciprocal function, identify the asymptotes or the value that makes the function undefined.
Plot some points both to the left and the right of the vertical asymptote. For the example, it would be appropriate to use the $x$values $1,$ $2,$ $2.5,$ $3.5,$ $4,$ and $5$ for this function.
$x$  $x−31 +2$  $f(x)=x−31 +2$ 

$1$  $1−31 +2$  $2.5$ 
$2$  $2−31 +2$  $3$ 
$2.5$  $2.5−31 +2$  $4$ 
$3.5$  $3.5−31 +2$  $0$ 
$4$  $4−31 +2$  $1$ 
$5$  $5−31 +2$  $1.5$ 
The points $(x,f(x))$ can be added to the coordinate plane to begin to see the behavior of $f.$
The graph can now be drawn by connecting the points with two smooth curves. They must approach but not intercept both the horizontal and vertical asymptotes.
Mark's school purchased a new math software program.
The program has a costs of $$360.$ Additionally, the school has to pay $$15$ per student who uses the software.
Verbal Expression  Algebraic Expression 

Initial cost  $360$ 
Cost for $n$ number of students  $15n$ 
Total cost  $15n+360$ 
Average cost $A$  $A=n15n+360 $ 
Write as a sum of fractions
Simplify quotient
Commutative Property of Addition
The domain of the function is all real numbers except $0$ and the range is all real numbers except $15.$ With this in mind, make a table of values. Be sure to include numbers on both sides of the vertical asymptote!
$n$  $n360 +15$  $A(n)=n360 +15$ 

$40$  $40360 +15$  $6$ 
$30$  $30360 +15$  $3$ 
$20$  $20360 +15$  $3$ 
$10$  $10360 +15$  $21$ 
$10$  $10360 +15$  $51$ 
$20$  $20360 +15$  $33$ 
$30$  $30360 +15$  $27$ 
$40$  $40360 +15$  $24$ 
Next, plot and connect the points from the table. Remember that the asymptotes are $x=0$ and $y=15$ and that the graph will have two branches.
Since $n$ represents the number of students using the software, it would not make sense for $n$ to be negative. Therefore, only the part of the graph in the first quadrant should be considered.
Notice that this transformation affects the vertical asymptote $x=h$ but not the horizontal one. The vertical and horizontal translations of the graph of a function $f$ can be summarized in a table.
Transformations of $y=x1 $  

$Vertical Translation y=x1 +k $

Translation up $k$ units, $k>0$ 
Translation down $k$ units, $k<0$  
$Horizontal Translation y=x−h1 $

Translation to the right $h$ units, $h>0$ 
Translation to the left $h$ units, $h<0$ 
Mark now has a rough idea about how to translate reciprocal functions. His class concluded that the asymptotes of a reciprocal function are also translated when the function is translated. Mark thought about how its domain and range were affected by this transformation. Consider the parent reciprocal function.
How will the domain and the range of the function $y=x1 $ change after the translation of its graph by $4$ units down and $3$ units to the left?
Domain: All real numbers except $3$
Range: All real numbers except $4$
When a function is translated, its graph keeps its original shape. Therefore, if it has any asymptotes, the asymptotes are translated in the same way.
To determine the domain and the range of the function after translating it, start by considering some possible transformations.
Transformations of $y=x1 $  

$Vertical Translation y=x1 +k $

Translation up $k$ units, $k>0$ 
Translation down $k$ units, $k<0$  
$Horizontal Translation y=x−h1 $

Translation to the right $h$ units, $h>0$ 
Translation to the left $h$ units, $h<0$ 
Note that if the graph of the function is translated, the asymptotes are also translated the same distance and direction. A combination of transformations can be applied to the same parent function. Consider the given translations again.
The second transformation is a horizontal translation $3$ units left.
Finally, look at the graph of the given function and its asymptotes alone.
As shown in the graph, the vertical asymptote is the line $x=3$ and the horizontal asymptote is the line $y=4.$ This information can be used to state the domain and range of the function.
The graph of the reciprocal function $f(x)=xa $ is shown in the coordinate plane. Observe how the graph changes as the value of $a$ changes.
Transformations of $y=x1 $  

$Vertical Stretch or Shrink y=xa $

Vertical stretch, $a>1$ 
Vertical shrink, $0<a<1$  
$Horizontal Stretch or Shrink y=bx1 $

Horizontal stretch, $0<b<1$ 
Horizontal shrink, $b>1$ 
The table below summarizes the different types of reflections.
Transformations of $y=x1 $  

Reflections  $In thexaxisy=(x1 ) $

$In theyaxisy=x1 $

Notice that $(x1 )=x1 .$ Reflecting the parent reciprocal function in the $x$axis and in the $y$axis will produce the same graph.
Pair of Functions  

Mark  $y=x1 $  $y=3(x1 )$  
Paulina  $y=x1 $  $y=x1 −6$  
Tiffaniqua  $y=x1 $  $y=x+51 $ 
Help them identify the transformations, asymptotes, domain, and range of the reciprocal functions.
Domain  Range  Vertical Asymptote  Horizontal Asymptote  

$y=x1 $  $R−{0}$  $R−{0}$  $x=0$  $y=0$ 
$y=x3 $  $R−{0}$  $R−{0}$  $x=0$  $y=0$ 
$y=x1 −6$  $R−{0}$  $R−{6}$  $x=0$  $y=6$ 
$y=x+51 $  $R−{5}$  $R−{0}$  $x=5$  $y=0$ 
When it is asked to apply two or more transformations to a function, it is often necessary to apply them in a certain order. Applying transformations in the order shown below will help to find a correct solution.
Consider two different composition of transformation of $y=x1 .$
$f(x)=x−25 +3$  

Domain  All real numbers except $2$ 
Range  All real numbers except $3$ 
Vertical Asymptote  $x=2$ 
Horizontal Asymptote  $y=3$ 
$x3x =3$
Multiply $term$ by $divisor$
Subtract down
$f(x)=x−25 +3$  

Domain  All real numbers except $2$ 
Range  All real numbers except $3$ 
Vertical Asymptote  $x=2$ 
Horizontal Asymptote  $y=3$ 