Post on 27-Mar-2020
7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
1
Goals: Graph rational functions Find the zeros and y-intercepts Identify and understand the asymptotes Match graphs to their equations Vocabulary: Function notation: Domain: Asymptote: Intercepts: Zero: Table of values: Graph: Sketch: Function notation: Function f with input x can be written in several ways:
f(x) = equation rule f: x equation rule f: f(x)= equation rule In all cases, when you input a value for x, the output is the y coordinate.
Ex. ( x1 , y1) is the same as ( x1 , f(x1) ) | | on the x axis on the y axix
Ex. A. For each of the following: a. find f(0), f(-1), f(2) b. Graph the function y = f (x)
i. f(x) = 3x +1 ii. f : x→− x2
7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
2
Graphing Simple Broken Rational Functions Definition of a Rational Function: A Rational Function is a function that has a variable, x, in the denominator.
For example: f : x→ 1x
f (x) = 2x − 5
f : x→ xx + 3
g :g(x) = x − 4x + 2
They all have a restriction on x. Therefore they have a restriction on their domain. That is why we call them broken. Ex. B
a. Fill the table of values for x = – 4 to x = 4 for g : x→ 3x + 2
x
y
b. Graph the function g At the “break” there is special behaviour. Let’s check out what happens when x is close to – 2, i.e. x =– 1.97, – 1.98, – 1.99, – 2.001,– 2.01,– 2.02,
x – 1.97
– 1.98 – 1.99 x = – 2 -2.001 – 2.01 – 2.02
y
error ! broken
This special behaviour is what defines an Asymptote.
7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
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Asymptotes An asymptote is a line which a graph comes closer and closer to, but never crosses.
Exploring Asymptotes: Activity:
Looking at the graph of f x( ) = 1x
using online graphing software (ex. geogebra), what do
we notice? Exploring graphing the broken rational function: Ex.C. Fill in the following table of values for each function. Plot the points, and graph the function.
a. f : x→ 1x
x
-100 -10 -2 -1 − 12
− 14
0 1
4
12
1 2 10 100
y
7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
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b. g : x→ 1x − 4
x
-10 -2 -1 0 1 2 3 3.5 4 4.5 5 6 10
y
c. h : x→ 3xx −1
x
-10 -2 -1 0 0.5 0.75 1 1.25 1.5 2 3 10
y
7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
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Finding the Asymptotes
Vertical Asymptotes: Vertical asymptotes occur at the restrictions on the domain. i.e. at the restriction on the x in the denominator. When x ≠ N , the equation of the vertical asymptote is x = N. Horizontal Asymptotes: The horizontal asymptote is a value that the output, y, will never reach, but will come closer and closer to. Here are two methods to find the horizontal asymptote.
Method 1: substitute in very large positive and negative values for x (extreme values) and see where the y value tends towards (comes closer to)
i.e. f(-10 000) f(-1000) f(1000) f(10 000)
or
Method 2: divide every term by x. Eliminate whatever will become insignificant as x becomes infinitely big.
Think: for example: 1x
becomes insignificant as x→∞ because 1∞
is very very small.
Also true for 5x
; as x→∞ , 5∞→ 0 or any number:
numberx
; as x→∞ , number
∞→ 0
Worked Example: Find the horizontal asymptote for f x( ) = x2x −1
by Method 1:
f −1000( ) = x2x −1
= −10002 −1000( )−1 =
−1000−2001
= 0.4997
close to 0.5 or 12
by Method 2:
x2x −1
⇒
xx
⎛⎝⎜
⎞⎠⎟
2xx
⎛⎝⎜
⎞⎠⎟ −
1x
⎛⎝⎜
⎞⎠⎟⇒ 1
2 − 1x
⎛⎝⎜
⎞⎠⎟
as x→∞ , 1∞→ 0 so
1
2 − 1x
⎛⎝⎜
⎞⎠⎟⇒ 12 − 0( ) =
12
the horizontal asymptote occurs at
y = 12
1
1-1-1
y = 1 2
f 1000( ) = x2x −1
= 10002 1000( )−1 =
10001999
= 0.5002
7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
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Ex. D Find the horizontal asymptotes. At first, use both methods. Then choose one method that you prefer.
a. f : x→ 3x +1
b. g(x) = −1x
c. f x( ) = x2x +1
d. h : x→ 42 − x
e. g :g x( ) = 3x +1x − 2
f. h x( ) = x +1x(x −1)
Graphing Rational Functions Ex. E Graph the first three functions from Ex. D. (first find and mark the asymptotes, then create a table of values).
a. f : x→ 3x +1
b. g(x) = −1x
c. f x( ) = x2x +1
7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
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The Shape of Things (Remember: graphs ‘hug’ asymptotes) How to sketch a graph given:
1. The x and y intercepts (if there are any) 2. The vertical asymptote(s) 3. The horizontal asymptote
Ex. Sketch the graph with the given parameters:
Using the shape of things to sketch graphs of rational functions (no tables of values)
1. Find and mark the vertical asymptote on the graph
Vertical asymptotes occur at the restriction on x in the denominator.
2. Find and mark the horizontal asymptote on the graph Use either method 1 or 2 to find the horizontal asymptote (ie. f(1000) & f(-1000) or divide by x and eliminate insignificants)
3. Find x and y intercepts and mark them on the graph.
for y intercepts: let x = 0 find y. ( 0 , ) for x intercepts: let y = 0 find x. ( , 0 )
7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
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Ex.F Graph each of the following using the sketching method:
a. f x( ) = xx +1
b. f : x→ 32x −1
7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
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Match the Graph
Look for key features: Intercepts and Asymptotes Ex. G Match the graph to its equation:
a. y = 2x −1
b. f (x) = xx −1
c. f : x→ 4x −1( )2
d. f : x→ − 3x −1
7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
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Noticing Patterns: Use graphing software to graph each set of functions on the same set of axis and then discuss the differences and similarities:
A B C D
y = 3x
y = 6x
y = −2x
y = − 0.4x
y = xx +1
y = 3x −1
y = 3x+1
y = 3x−1
f x( ) = xx −1
f x( ) = 2xx −1
f x( ) = 4xx −1
f x( ) = 2x +1x −1
f : x→ xx +1( )2
f : x→ 3x2
f : x→ 6x2 + x
Practice Exercises: P108
Characteristics of broken rational functions
Example 1
Which numbers are not included in the definition
set?
a) 32
4:
�xxf �
b) 1
4:
2 �xxg �
Sketch the function graph of g.
Solution:
a) For x=1,5 the denominatorof the function
term ist zero. The number 1,5 can therefore not
be includes in the drfinition set.
b) As x2 is greater than or equal to 0, whatever
number is entered fpr the variable, x2+1
cannot be equal to 0 for any inserted value.
x -4 -2 -1 0 1 2 4
g(x)17
4 5
4 2 4 2 5
4 17
4
As no number can be excluded from the domain
no vertical asymptotes exist either
.
Example 2
The function 43
2:
�x
xxf � is given.
a) Which number cannot be included in the
domain? Indicate Df. Create a table of values
and draw te function graph
b) Indicate the equations of both asymptotes and
give reasons for them.
c) What percentage of the horizontal asymptote
is f(1000)?
Solution:
a) The denominator 3x+4 can equal 0 when
4
3��x . The number
3
4� can therefore not be
included in the domain.
}3
4{\ ��QDf
b) Verical asymptote: 3
4��x
For relatively high x values the y-values
approach the number 3
2. The summand 4 in
the denomiator has a minor influence on the y-
value. The horizontal asymptote is therefor
described by 3
2�y
If one divides numerator and
denominator by x, one receives
the equation equivalent
(for x � 0)
x
xf4
3
2)(
��
By these it can easily be shown
that the y values of absolute
great x approximate
3
2
c) 3
2%87,99...6657,0)1000( off ��
x -100 -5 -4 -3 -2 -1,5 -1 0 1 2 5 100
y 0,68 0,9 1 1,2 2 6 -2 0 0,3 0,4 0,5 0,66
2 Which number(s) cannot be included in the
domain? Indicate a domain for each one.
a) x
xf32
4:
�� b)
xxf
2
3: �
c) 2
4:
xxf � d)
9
2:
2 �x
xxf �
e) 34: �zxf � f) )3()3(
2:
���
xx
xxf �
g) 52
13:
��
x
xxf � h)
tt
ttf
21
3:
���
�
i) xx
xf�2
1: � k)
4
12:
�ssf �
3 Exercises
Mixed (salad of) solutions
for 2:
For the following functions you cannot calculate
the restrictions on the domain directly. Why not?
Try to find the numbers using systematic
sampling.
a) 15,2
3:
2 ��xxf �
b) 8
3:
3 �x
xxg �
c) 483
27:
4
2
���
x
xxxh �
i.e. what are the restrictions on x ?
be included in the definition set (domain).
2for x, x + 1 will never become zero. Therefore, there are no definition
holes (no restrictions).
the function graph.
4
3
x
(x) =
(x) =
P 108
7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
11
P109
Characteristics of broken rational functions
4 �
Use a function plotter to draw the graphs of the
functions. Indicate the equations of horizontal
and ertical asymptotes, if they exist. Describe
how you could also have found these without th
graph by using the function rule.
a) x
xf3
1: � b)
13
2:
�xxf �
c) 13
2:
�xx
xf � d) s
sf�5,1
1: �
e) t
ttf
�5,1: �
f) )2()2(
1:
�� xxxf �
5
Given each function rule, ansider the graph
includes asymptotes and, if relevant, indicate the
equations of the asymptotes. Draw the function
graph (without using a table of values or a
function plotter).
a) x
xf41
1:
�� b)
)1(2
3:
�xxg �
c) 12
:�xx
xh � d) x
xk5,2
: �
e) x
xp5,02
1:
��
f) )3(
2:
��xx
xq �
6
Draw both graphs for the functions x
xf4
: �
and 2
4:
xxg � in a common coordinate
system. Describe the similarities and differences
of the two graphs in words.
7
a) Draw the graph of the function 3
1:
�xxf � .
Describe how the graph would change if the
number 3 was replaced with
- a larger number
- a smaller number
b) What would change in the graph of f if in the
numerator the variable x was included instead
of the number 1?
c) Using a function plotter draw the graphs of the
functions 3
:1 �xx
xf � ;32
:2 �xx
xf � and
33:3 �x
xxf � in a common coordinate
system and describe the differences between
the graphs.
8
Draw the graphs of the two functions x
xf3
: �
and xxg3
4
3
20: �� in a common coordinate
system. Read off the intersections of each graph
as precisely as possible. Determine the
approximate value the surface area of the areas
enclosed by the intersections of the two graphs.
9
Which numbers must be excluded from the
definition set? Indicate the definition set and
sketch the function graphs.
a) 2)2(
4:
�xxf �
b) )3(
1:
xxxg
��
10
Indicate each time a broken rational function
which has the line with the given equation as
asymptote.
a) x = 1 b) y = -2
c) x = 0 ; the graph should also go through the
point (2�4).
11
Indicate a broken rational function for which the
graph does not enter the yellow area.
vertical
the graph by using the function rule (equation).
consider if the graph
( x ) =
(Write the function rule for a graph).
For each, write the rule for a broken rational function
P 109
Characteristics of broken rational functions
4 �
Use a function plotter to draw the graphs of the
functions. Indicate the equations of horizontal
and ertical asymptotes, if they exist. Describe
how you could also have found these without th
graph by using the function rule.
a) x
xf3
1: � b)
13
2:
�xxf �
c) 13
2:
�xx
xf � d) s
sf�5,1
1: �
e) t
ttf
�5,1: �
f) )2()2(
1:
�� xxxf �
5
Given each function rule, ansider the graph
includes asymptotes and, if relevant, indicate the
equations of the asymptotes. Draw the function
graph (without using a table of values or a
function plotter).
a) x
xf41
1:
�� b)
)1(2
3:
�xxg �
c) 12
:�xx
xh � d) x
xk5,2
: �
e) x
xp5,02
1:
��
f) )3(
2:
��xx
xq �
6
Draw both graphs for the functions x
xf4
: �
and 2
4:
xxg � in a common coordinate
system. Describe the similarities and differences
of the two graphs in words.
7
a) Draw the graph of the function 3
1:
�xxf � .
Describe how the graph would change if the
number 3 was replaced with
- a larger number
- a smaller number
b) What would change in the graph of f if in the
numerator the variable x was included instead
of the number 1?
c) Using a function plotter draw the graphs of the
functions 3
:1 �xx
xf � ;32
:2 �xx
xf � and
33:3 �x
xxf � in a common coordinate
system and describe the differences between
the graphs.
8
Draw the graphs of the two functions x
xf3
: �
and xxg3
4
3
20: �� in a common coordinate
system. Read off the intersections of each graph
as precisely as possible. Determine the
approximate value the surface area of the areas
enclosed by the intersections of the two graphs.
9
Which numbers must be excluded from the
definition set? Indicate the definition set and
sketch the function graphs.
a) 2)2(
4:
�xxf �
b) )3(
1:
xxxg
��
10
Indicate each time a broken rational function
which has the line with the given equation as
asymptote.
a) x = 1 b) y = -2
c) x = 0 ; the graph should also go through the
point (2�4).
11
Indicate a broken rational function for which the
graph does not enter the yellow area.
vertical
the graph by using the function rule (equation).
consider if the graph
( x ) =
(Write the function rule for a graph).
For each, write the rule for a broken rational function
P 109
Characteristics of broken rational functions
4 �
Use a function plotter to draw the graphs of the
functions. Indicate the equations of horizontal
and ertical asymptotes, if they exist. Describe
how you could also have found these without th
graph by using the function rule.
a) x
xf3
1: � b)
13
2:
�xxf �
c) 13
2:
�xx
xf � d) s
sf�5,1
1: �
e) t
ttf
�5,1: �
f) )2()2(
1:
�� xxxf �
5
Given each function rule, ansider the graph
includes asymptotes and, if relevant, indicate the
equations of the asymptotes. Draw the function
graph (without using a table of values or a
function plotter).
a) x
xf41
1:
�� b)
)1(2
3:
�xxg �
c) 12
:�xx
xh � d) x
xk5,2
: �
e) x
xp5,02
1:
��
f) )3(
2:
��xx
xq �
6
Draw both graphs for the functions x
xf4
: �
and 2
4:
xxg � in a common coordinate
system. Describe the similarities and differences
of the two graphs in words.
7
a) Draw the graph of the function 3
1:
�xxf � .
Describe how the graph would change if the
number 3 was replaced with
- a larger number
- a smaller number
b) What would change in the graph of f if in the
numerator the variable x was included instead
of the number 1?
c) Using a function plotter draw the graphs of the
functions 3
:1 �xx
xf � ;32
:2 �xx
xf � and
33:3 �x
xxf � in a common coordinate
system and describe the differences between
the graphs.
8
Draw the graphs of the two functions x
xf3
: �
and xxg3
4
3
20: �� in a common coordinate
system. Read off the intersections of each graph
as precisely as possible. Determine the
approximate value the surface area of the areas
enclosed by the intersections of the two graphs.
9
Which numbers must be excluded from the
definition set? Indicate the definition set and
sketch the function graphs.
a) 2)2(
4:
�xxf �
b) )3(
1:
xxxg
��
10
Indicate each time a broken rational function
which has the line with the given equation as
asymptote.
a) x = 1 b) y = -2
c) x = 0 ; the graph should also go through the
point (2�4).
11
Indicate a broken rational function for which the
graph does not enter the yellow area.
vertical
the graph by using the function rule (equation).
consider if the graph
( x ) =
(Write the function rule for a graph).
For each, write the rule for a broken rational function
P 109
7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
12
P110
P126
Characteristics of broken rational functions
12
The function graphs shown should be matched to
the function terms given. In the order a) to h) the
correctly matched letters result in a solution
word. But first you have to >break> another code.
13
Indicate the broken rational functions for which
the graphs go through the point P (3�2).
14
Draw the graph of the function 12
2:
�xxf � ,
How would the graph change if the 2 was
replaced by a larger number? By a smaller
number?
15
The following functions each have a restriction
on the domain. The graphs of the function,
however, show four differenz behaviours near
these restrictions. Draw the function geaphs with
a function plotter and describe the four different
behaviours.
a)x
xf�3
2:1 � b)
12
3:2 �xxf �
c)23
4:
xxf � d)
1:4 �x
xxf �
e) 25)1(
2:
xxf
��
� f)
5,1
1:6 �xxf �
g)1
2:7 �x
xxf � h)
28
1:
xtf �
Could you determine which of these behaviours
will occur just by considering the function rule?
G16
The half-lines w1 and w2 are the bisectors of �
and �. The angle � is selected so that it is halved
by the line g.
a) Transfer the figure for �=40° into the notebook
(the points A and B are at random distances
away from S).
b) At wich angle � do the points A, S and B lie an
a line.
c) How should � be selected so that B ans S
create a line with w1? Give reasons.
3
3:
�xx
xf � |M 23: �xxf � |J
2)3(
4:
�xxf � |C
)2)(6(
1:
�� xxxf � |B
3
3:
�xxf � |W
3
)3(:
��
x
xxxf � |B
3(
2:
��x
xf � |S 2)3(: �xxf � |F
rulesa
the intersection of asymptotes occurs at the point
P( 3 | 2 )
t
2)
P 110
Characteristics of broken rational functions
12
The function graphs shown should be matched to
the function terms given. In the order a) to h) the
correctly matched letters result in a solution
word. But first you have to >break> another code.
13
Indicate the broken rational functions for which
the graphs go through the point P (3�2).
14
Draw the graph of the function 12
2:
�xxf � ,
How would the graph change if the 2 was
replaced by a larger number? By a smaller
number?
15
The following functions each have a restriction
on the domain. The graphs of the function,
however, show four differenz behaviours near
these restrictions. Draw the function geaphs with
a function plotter and describe the four different
behaviours.
a)x
xf�3
2:1 � b)
12
3:2 �xxf �
c)23
4:
xxf � d)
1:4 �x
xxf �
e) 25)1(
2:
xxf
��
� f)
5,1
1:6 �xxf �
g)1
2:7 �x
xxf � h)
28
1:
xtf �
Could you determine which of these behaviours
will occur just by considering the function rule?
G16
The half-lines w1 and w2 are the bisectors of �
and �. The angle � is selected so that it is halved
by the line g.
a) Transfer the figure for �=40° into the notebook
(the points A and B are at random distances
away from S).
b) At wich angle � do the points A, S and B lie an
a line.
c) How should � be selected so that B ans S
create a line with w1? Give reasons.
3
3:
�xx
xf � |M 23: �xxf � |J
2)3(
4:
�xxf � |C
)2)(6(
1:
�� xxxf � |B
3
3:
�xxf � |W
3
)3(:
��
x
xxxf � |B
3(
2:
��x
xf � |S 2)3(: �xxf � |F
rulesa
the intersection of asymptotes occurs at the point
P( 3 | 2 )
t
2)
P 110
Characteristics of broken rational functions
12
The function graphs shown should be matched to
the function terms given. In the order a) to h) the
correctly matched letters result in a solution
word. But first you have to >break> another code.
13
Indicate the broken rational functions for which
the graphs go through the point P (3�2).
14
Draw the graph of the function 12
2:
�xxf � ,
How would the graph change if the 2 was
replaced by a larger number? By a smaller
number?
15
The following functions each have a restriction
on the domain. The graphs of the function,
however, show four differenz behaviours near
these restrictions. Draw the function geaphs with
a function plotter and describe the four different
behaviours.
a)x
xf�3
2:1 � b)
12
3:2 �xxf �
c)23
4:
xxf � d)
1:4 �x
xxf �
e) 25)1(
2:
xxf
��
� f)
5,1
1:6 �xxf �
g)1
2:7 �x
xxf � h)
28
1:
xtf �
Could you determine which of these behaviours
will occur just by considering the function rule?
G16
The half-lines w1 and w2 are the bisectors of �
and �. The angle � is selected so that it is halved
by the line g.
a) Transfer the figure for �=40° into the notebook
(the points A and B are at random distances
away from S).
b) At wich angle � do the points A, S and B lie an
a line.
c) How should � be selected so that B ans S
create a line with w1? Give reasons.
3
3:
�xx
xf � |M 23: �xxf � |J
2)3(
4:
�xxf � |C
)2)(6(
1:
�� xxxf � |B
3
3:
�xxf � |W
3
)3(:
��
x
xxxf � |B
3(
2:
��x
xf � |S 2)3(: �xxf � |F
rulesa
the intersection of asymptotes occurs at the point
P( 3 | 2 )
t
2)
P 110
Fractional equations���S���
11
a) If you add the same number of the
numerator and denominator of the fraction
12
5 you get the fraction
5
4 . What is the
number?
b) A fraction when cancelled gives
3
2 . If you
add the number 5 to the uncancelled
numerator and denominator you get the
fraction .
7
5.
c) What is the uncancelled fraction?
12
Find the intersections of the graphs of
1
2:
2
��
x
xxf � and 3: �xxg �
mathematically.
Sketch both function graphs on the same axes.
13
Solve the following formulas for the variables
in the brackets.
a) )(VV
m��
b) )()(2
1chcaA ���
c) )(2
1h
h
caB
���
d) )(2
2
1
2
1 tt
t
F
F�
e) )(2
1 2 ggtF � f) )(
3b
b
bat
���
14
Karl learned the formula
v
sttvs
t
sv ���� ,,
by heart for his homework.
Claudia says: I just need to know the formula
t
sv � . I can get both the other equations just
by transformation.
Discuss the advantages and disadvantages of
KarlOs and ClaudiaOs approaches. How would
you do it?
15
a) Is there a fraction with the same value as the
number 0.875 whose denominator is 3
greater than the numerator?
b) For which fractional equation for 0.875 do
the numerator and denominator add up to
300?
16
Write a fractional equation in which the
variable comes up at least twice and the
number 3.5 is the solution. Give the equation
to your neighbour to solve.
17
Describe the difference between the graphs of
the function
4:
xxf � and
xxg
4: � in
words. Calculate the coordinates of the
intersection of both graphs.
18
The Wittke family is going on holiday by car.
Ramona has calculated that they first need to
drive 645 km on the Autobahn and then 90 km
on the regional roads. Mr. Wittke says that
they can drive twice as fast on the Autobahn as
on the regional roads. Mrs. Wittke then says:
YSo we will need
2
17 hours for the journey.Z
What speeds does she base her calculation on?
19
A swimming pool is beeing emptied using two
pumps. One of the pumps on its own would
need 3 hours, the other 2 hours.
a) How long do both pumps need when they
are used at the same time?
b) How long would a third pump need alone if
the tree pumps together would manage it in
one hour?
c) Check if it is possible to use a third pump so
that three pumps together would manage the
job in 45 minutes.
20
a) The equation has two
solutions which can be quite easily
recognised without transformations. What
are they?
3)3)(1( ���� xxx
b) Solve the equations and 11��x
2
1
2
1
��
��
xx
x
c) What transformations can you use to get the
equation in b) from the one in a)?
d) Compare the solutions of the three equations
and describe the effect of the
transformations you did.
7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)
13
Extra Practice Graphing: Graph each of the following functions. Indicate the intercepts and equations for the asymptotes in each case.
a. y = 2x
b. y = −1x − 2
c. f x( ) = xx − 3
d. f x( ) = −x2x + 4
e. f x( ) = 2xx +1
f. f x( ) = 0.5x +1x +1
g. f : x→ −x + 3x − 3
h. f : x→ 2x +1x
i. h :h(x) = 3x(x − 2)
Answers: S.108 S.109