Chapter 2 Analysis of Graphs of Functions - Cerritos...

Chapter 2 Analysis of Graphs of Functions

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Chapter 2

Analysis of Graphs of Functions

Covered in this Chapter:

2.1 Graphs of Basic Functions and their Domain and Range. Odd, Even Functions, and

their Symmetry.

2.2 Translations of Graphs: Vertical and Horizontal Shift, Stretching and Compressing,

and Reflections.

2.3 Graphs of Absolute Value Functions. Equations, Inequalities, and Applications.

2.4 Piecewise Functions, Greatest Integer Function, and their Graphs.

2.5 Algebra of Functions and their Composition. Difference Quotient.

► Answers to Exercises.


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2.1 Graphs of Functions

Definitions:

Continuous Functions:

A function is called continuous on its domain if it has no tears, gaps, or holes. That is, if it can be drawn

without lifting your pencil when you draw it on paper.

Increasing and Decreasing Functions:

2 1 2 1

2 1 2 1

A function is called increasing on an interval if

in

A function is called decreasing on an interval if

in

f I

f x f x whenever x x I

f I

f x f x whenever x x I

Constant Functions:

2 1 2 1 2 1

A function is called constant on an interval if

in (or in general, whenever )

f I

f x f x whenever x x I x x

Symmetry of Functions:

Even Functions

A function is called an EVEN function if it satisfies the condition for all in its domain.

i.e. If the point , is on the graph of , then the point , must be also on its gra

f f x f x x

x y f x x y

ph.

An EVEN function is symmetric about the -axis.y

8

4

2

6

10

x

y

2 4 6 8

2

4

6

8

6 248 10

10

10 0

5y

2x

A

0 a1x b c d

D

C

By

x

2f x

1f x

y f x

Increasing Increasing

Decreasing

Section 2.1 Graphs of Functions

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Examples of Even Functions

Odd Functions

A function is called an ODD function if it satisfies the condition for all in its domain.

i.e. If the point , is on the graph of , then the point , must be also on its graph.

f f x f x x

x y f x x y

An ODD function is symmetric about the origin.

Examples of Odd Functions

Tests for Symmetry:

Symmetry about the y-axis (even functions):

Replace with in ( ). If ( ) ( ), then the graph is symmetric about the -axis.x x f x f x f x y

Symmetry about the origin (odd functions):

Replace with in ( ). If ( ) ( ), then the graph is symmetric about the origin.x x f x f x f x

Symmetry about the x-axis:

Replace with in the equation. If you get the same equation, then the graph is symmetric about the

-axis. Remember that graphs symmetric about the -axis are not functions. See the example below.

y y

x x

0

f x f x

xx

x

y

x

y

0

f x

f xx

x


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Example 1: Decide whether the function is odd, even or neither. 4 22 5 10f x x x

Solution: First replace x with –x in the given function.

4 2

4 2

4 2

2 5 10

( ) 2( ) 5( ) 10

( ) 2 5 10 ( )

( ) ( )

The given function is EVEN.

(i.e. The function is symmetric about the -axis.)

f x x x

f x x x

f x x x f x

f x f x

y

Example 2: Decide whether the function is odd, even or neither. 35 4 3f x x x


3

3

3

3

5 4 3

( ) 5( ) 4( ) 3

( ) 5 4 3

( ) (5 4 3)

( ) ( ) nor ( )

The given function is NEITHER odd nor even.

f x x x

f x x x

f x x x

f x x x

f x f x f x

Example 3: Decide whether the function is odd, even or neither. 3

2

2 5

4 1

x xf x

x


3

2

3

2

3

2

3

2

2 5

4 1

2( ) 5( )

4( ) 1

2 5

4 1

2 5

4 1

( ) ( )

The given function is ODD.

(i.e. The function is symmetric about the origin.)

x xf x

x

x xf x

x

x xf x

x

x xf x

x

f x f x


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3

Domain: or ( , )

Range: or ( , )

Odd Function

Symmetric about the origin

y x

Basic Functions and their Graphs

Identity Function Absolute Value Function

Domain: or ( , )

Range: or ( , )

Odd Function


y x

Domain: or ( , )

Range: y 0 or [0, )

Even Function

Symmetric about the -axis

y x

y

Quadratic Function Square Root Function

2

Domain: or ( , )

Range: y 0 or [0, )

Even Function

Symmetric about the -axis

y x

y

Domain: 0 or [0, )

Range: y 0 or [0, )

Function is neither odd nor even

y x

x

Cubic Function


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3

Domain: or ( , )

Range: or ( , )

Odd function


y x

Cube Root Function

********************************************************

Exercises 2.1

In exercises 1 – 12, decide whether the function is odd, even, or neither and state whether its graph is

symmetric about the origin, y-axis, or neither.

1) 3f x x 2) 4f x x x

3) 4 22 5 3f x x x 4) 34 3 1f x x x

5) 5 33 2 6f x x x x 6) 3 22 4 5 1f x x x x

7) 2

2

3 1

2

xf x

x

8)

3

2

4

3

x xf x

x

9) 32 3

1

x xf x

x

10)

5 3

3

2 4

3 5

x xf x

x x

11) 4 2

3

3 5 1

2 5

x xf x

x x

12)

2

3

4

5 2

xf x

x x


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In exercises 13 – 24, decide whether the graph is symmetric about the x-axis, y-axis, or the origin.

13) 14) 15)

16) 17) 18)

19) 20) 21)

22) 23) 24)


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2.2 Translations of Graphs

In this section, translations of graphs will be discussed. We will be using a screen produced by a TI 84+

showing an original function drawn in a thick line and some other functions that were produced from that

original function. We should be able to come up with a conclusion from all of these screens.

Vertical Shifts

The above two screens show that if ( ) is a function and is a positive number, then

the graph of ( ) is exactly the same as the graph of ( ) but shifted down units.

f x c

f x c f x c


the graph of ( ) is exactly the same as the graph of ( ) but shifted up units.

f x c

f x c f x c

Horizontal Shifts


the graph of ( ) is exactly the same as the graph of ( ) but shifted to the right units.

f x c

f x c f x c

Section 2.2 Translations of Graphs

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the graph of ( ) is exactly the same as the graph of ( ) but shifted to the left units.

f x c

f x c f x c

Vertical Stretching

The above screens show that if ( ) is a function and is a positive number greater than 1 ( 1), then

the graph of [ ( )] is exactly the same as the graph of ( ) but vertically stretched by a facto

f x c c

cf x f x

r of c

Vertical Shrinking

The above screens show that if ( ) is a function and is a positive number such that 0 1, then

the graph of [ ( )] is exactly the same as the graph of ( ) but vertically shrunk by a factor of .

f x c c

cf x f x c


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Horizontal Shrinking

The above screens show that if ( ) is a function and is a positive number greater than 1 ( 1), then

the graph of ( ) is exactly the same as the graph of ( ) but horizontally shrunk by a factor o

f x c c

f cx f x

f .c

Horizontal Stretching

The above screens show that if ( ) is a function and is a positive number such that 0 1, then

the graph of ( ) is exactly the same as the graph of ( ) but horizontally stretched by a factor of

f x c c

f cx f x

.c

Reflection About the x-axis

The above screens show that if ( ) is a function, then the graph of [ ( )]

is exactly the same as the graph of ( ) but reflected about the -axis.

f x f x

f x x


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Reflection About the y-axis

The above screens show that if ( ) is a function, then the graph of ( )

is exactly the same as the graph of ( ) but reflected about the -axis.

f x f x

f x y

Now, that we know all operations that can be performed on a given graph, we should be able to answer

the following examples.

Example 1: The graph of the function ( )f x x is shifted to the right 3 units, then vertically stretched by

a factor of 4, then reflected about the y-axis, and finally shifted down 5 units. Write the final translated

function h(x) and give its graph.

Solution:

( )f x x Shifted to the right

3 units3x

Vertically stretched bya factor of 4 4 3x

Reflected about the -axis

4 3y

x Shifted down

5 units 4 3 5 ( )x h x


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Example 2: The graph of the function ( )f x x is shifted to the left 4 units, then vertically shrunk by a

factor of 1/2, then reflected about the x-axis, and finally shifted up 3 units. Write the final translated

function h(x) and give its graph.

Solution:

( )f x x Shifted to the left4 units

4x Vertically shrunk by

a factor of 1/2

1 42

x

Reflected about

the -axis

14

2x x Shifted up

3 units4 3

1 ( )

2x h x

Example 3: The graph of the function f x x is obtained by performing the following transformations

on the graph of the function g x in the given order: shifted horizontally 5 units to the left, then vertically

stretched by a factor of 4, reflected about the y-axis, and finally shifted downward 3 units. Find the rule of

the function g x .

Solution: We will show two methods to solve these types of problems.

Method 1: Following the exact steps as given and equating the final answer to f x .

vertically stretchedshifted to the left reflected about5 units by a factor of 4 the -axis5 4 5 4 5yg x g x g x g x

shifted down3 units4 5 4 5 3 ; solve for g x g x f x x g x

4 5 3 ; add 3 to both sides of the equationg x x

4 5 3; multiply both sides of the equation by 1/4g x x


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15 3 ; reflect about the -axis

4

1 15 3 3 ; shift to the right 5 units to obtain

4 4

15 3

4

g x x y

g x x x g x

g x x

Note: The absolute value of any negative number is equal to the absolute value of the number itself.

x x

Method 2: The function g x is shifted to the left 5 units, vertically stretched by a factor of 4, reflected

about the y-axis, and finally shifted down 3 units to obtain the function f x x .

Perform the opposite of each transformation in reverse order on f x to obtain the rule of the function

g x . The function f x x is shifted up 3 units, reflected about the y-axis, vertically shrunk by a factor

of 1/4, and finally shifted to the right 5 units.

Shift f x x up 3 units: 13 3f x x f x

Reflect 1 3f x x about the y-axis: 1 23 3f x x x f x

Vertically shrink 2 3f x x by a factor of1

4: 2 3

1 13

4 4f x x f x

Shift 3

13

4f x x to the right 5 units: 3

15 5 3

4f x x g x

1

5 34

g x x


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Exercises 2.2

In exercises 1 – 8, write the function g whose graph can be obtained from the graph of the function f by

performing the transformations in the given order. Graph the function g.

21) 3;f x x shift the graph horizontally 2 units to the right and then vertically downward 5 units.

22) 1;f x x shift the graph horizontally 3 units to the left and then vertically upward 4 units.

23) 2 1;f x x x reflect the graph about the y-axis, then shift it vertically downward 3 units.

24) 4 4;f x x x reflect the graph about the y-axis, then shift it vertically downward 1 unit.

25) 2 1;f x x x reflect the graph about the x-axis, then shift it vertically upward 2 units.

26) 4 4;f x x x reflect the graph about the x-axis, then shift it vertically upward 3 units.

7) ;f x x shift the graph horizontally to the right 4 units, stretch it vertically by a factor of 3, reflect

the graph about the y-axis, then shift it vertically upward 5 units.

8) ;f x x shift the graph horizontally to the left 3 units, stretch it vertically by a factor of 2, reflect

the graph about the y-axis, then shift it vertically downward 1 unit.

In exercises 9 – 14, use the graph of the function g in the

given figure to sketch the graph of the function f.

9) 3 10) 2

11) 2 12) 0.5

13) 14)

f x g x f x g x

f x g x f x g x

f x g x f x g x


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In exercises 15 – 20, use the graph of the function g in the

given figure to sketch the graph of the function f.

15) 1 16) 3

17) 2 18) 0.5

19) 2 20) 1

f x g x f x g x

f x g x f x g x

f x g x f x g x

21) The graph of the function 2f x x is obtained by performing the following transformations on the

graph of the function g x in the given order: shifted horizontally 2 units to the right, then vertically

shrunk by a factor of 0.5, reflected about the x-axis, and finally shifted downward 3 units. Find the

rule of the function g x .

22) The graph of the function 2f x x is obtained by performing the following transformations on the

graph of the function g x in the given order: shifted horizontally 3 units to the left, then vertically

stretched by a factor of 2, reflected about the y-axis, and finally shifted upward 5 units. Find the rule

of the function g x .

23) The graph of the function f x x is obtained by performing the following transformations on the

graph of the function g x in the given order: shifted horizontally 1 unit to the right, then vertically

shrunk by a factor of 0.5, reflected about the y-axis, and finally shifted downward 2 units. Find the


24) The graph of the function f x x is obtained by performing the following transformations on the

graph of the function g x in the given order: shifted horizontally 4 units to the left, then vertically

shrunk by a factor of 0.25, reflected about the x-axis, and finally shifted downward 3 units. Find the



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2.3 Absolute Equalities and Inequalities

Absolute Value of a Function:

if 0

if 0

f x f xf x

f x f x

The above notation is simply saying that when the graph is zero or positive, leave it as it is. If the graph is

negative then reflect it about the x-axis to make it positive.

Example 1: Given that 4 20.1 2 3f x x x , use your graphing calculator to graph y f x .

Solution: The first screen below shows the graph of the function f(x). The second screen shows both the

graph of the function in thick lines and the graph of its absolute together. The last screen shows only the

graph of the absolute of the function.

Absolute Equations:

if 0

if 0

x xx

x x

Absolute value of x measures the distance from x to zero. Remember that distance must be non-negative.

Therefore 7 7, 1 1, 0 0, 5 5, 10 10, and so on since the distance from any number to zero

is the positive case of that number.

Example 2: Solve the absolute equation 3x .

Solution: The problem is simply saying that if the distance from x to zero is 3 units, what is x? We know

that the distance must be non-negative. If we want to measure 3 units from zero, we have two ways to do

so. Either we count 3 units to the right (this case we get +3) or we can count 3 units to the left (this case

we get -3). This means we have two answers for x, +3 and -3. The figure below shows how we got these

two answers.

3x

3 units3 units

0 1 2 33 2 1

Section 2.3 Absolute Equations and Inequalities

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To generalize the method of solving absolute equations, we have the following rules:

If where is a positive number ( 0) then,

or

If where is a negative number ( 0) then

the equation has no sol

ax b c c c

ax b c ax b c

ax b c c c

ution

If 0 then, 0ax b ax b

Example 3: Solve the absolute equation 3 5x .

Solution: First we need to apply the rule above and remove the absolute symbol by writing two

equations.

3 5

5 3

2

x

x

x

or

3 5

5 3

8

x

x

x

; 2, 8x

Example 4: Solve the absolute equation 2 7 6 5x .

Solution: We have to isolate the absolute expression before removing the absolute symbol. Therefore the

equivalent equation is 2 7 11x . Now we should be able to write two equations and solve the problem.

2 7 11

2 18

9

x

x

x

or

2 7 11

2 4

2

x

x

x

; 2,9x

Example 5: Solve the absolute equation 2 7 8 3x .

Solution: We have to isolate the absolute expression before removing the absolute symbol. Therefore the

equivalent equation is 2 7 5x . But, we know that absolute value must be non-negative all the time.

Therefore, the given absolute equation has no solution.

Example 6: Solve the absolute equation 2 10 0x .

Solution: In this type of problems when you see zero on one side of the equation, simply remove the

absolute symbol and solve the equation as usual.

2 10 0

2 10 0

2 10

5

x

x

x

x


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Solving Absolute Equations of the Type ax b cx d

If , then

or

ax b cx d

ax b cx d ax b cx d

Example 7: Solve the absolute equation 2 7 3 5x x .

Solution: Follow the above rule and write two equations to solve the problem.

2 7 3 5

12

12

x x

x

x

2 7 3 5

2 7 3 5

25 2,

5

x x

x x

x x

212,

5x

Example 8: Solve the absolute equation 3 5 3 2x x .

Solution: Write two equations and solve the problem.

3 5 3 2

3 3 2 5

0 7

No Solution

x x

x x

3 5 3 2

3 5 3 2

6 3

1

2

x x

x x

x

x

1

2x

Note: In the last example, don’t write “the answer is no solution or -1/2”. Just write -1/2

Solving Absolute Inequalities

Solving Inequalities of the Form ax b c

If where is a positive number 0 , then

and

If we write the above as continued inequalities, we can write the inequ

ax b c c c

ax b c ax b c

ality as follows:

c ax b c

If where c is a negative number 0 , then

the inequality has no solution

If 0, then 0

ax b c c

ax b ax b

Note: In the above rule, due to space limitation, we used the symbol of inequality less than or equal .

The less than symbol also applies to this rule.


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The figure below shows why we call the above inequality the AND case. Notice that the interval of the

solution is one continuous interval. If x is in the interval (-3, 3), we can see that x is less than 3 AND x is

greater than -3.

Example 9: Solve the absolute inequality.

3 5x

Solution: First we need to apply the rule above

and remove the absolute symbol by writing

continued inequalities as follows:

5 3 5

5 3 3 3 5 3

8 2

Set Notation: { | 8 2}

Interval Notation: [-8,2]

x

x

x

x x


2 7 6 5x

Solution: We have to isolate the absolute

expression before removing the absolute symbol.

Therefore the equivalent inequality is

2 7 11x . Now we should be able to write the

inequality as continued inequalities and solve the

problem.

11 2 7 11

11 7 2 7 7 11 7

4 2 18

2 9


Interval Notation: ( 2,9)

x

x

x

x

x x


2 7 8 3x

Solution: We have to isolate the absolute

expression before removing the absolute symbol.

Therefore the equivalent inequality is

2 7 5x .

But we know that absolute value must be non-

negative all the time. Therefore, the given

absolute inequality has no solution.


2 10 0x

Solution: In these types of problems when you

see that the inequality expression is less than or

equal to zero, remove the absolute symbol,

replace the inequality symbol with an equal sign

and solve the equation.

2 10 0

2 10 0

2 10

5

x

x

x

x

3x

0 1 2 33 2 1

less than3 units

less than3 units


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Solving Inequalities of the Form ax b c

If where is a positive number 0 , then

or

If where is a non-positive number 0 , then

the

ax b c c c

ax b c ax b c

ax b c c c

solution is the set of all real numbers

If 0, then the solution is all real numbers except

that is: , ,

bax b x

a

b b

a a

Note: In the above rule, due to space limitation, we used the symbol of inequality greater than or equal

. The greater than symbol also applies to this rule.

The figure below shows why we call the above inequality the OR case. Notice that the interval of the

solution is two separate intervals. If x is in the interval , 3 then x is less than -3. If x is in the

interval 3, then x is greater than 3. Since x cannot be in two places at the same time, we have to say

that x is less than -3 OR x is greater than 3.

Example 13: Solve the absolute inequality 3 5x

Solution: First we need to apply the rule above and remove the absolute symbol by writing the absolute

inequality as two separate inequalities.

3 5

3 3 5 3

8

x

x

x

or

3 5

3 3 5 3

2

x

x

x

Set Notation: { | 8 or 2}

Interval Notation: (- ,-8] [2, )

x x x

Example 14: Solve the absolute inequality 2 7 8 3x

Solution: We have to isolate the absolute expression before removing the absolute symbol. Therefore, the

equivalent inequality is 2 7 5x .

But, we know that absolute value must be non-negative all the time. Therefore, the solution to the given

absolute inequality is the set of all real numbers.

3x

greater than

0 1 2 33 2 1

3 units

greater than

3 units


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Example 15: Solve the absolute inequality 2 7 6 5x

Solution: We have to isolate the absolute expression before removing the absolute symbol. Therefore, the

equivalent inequality is 2 7 11x . Now we should be able to write the inequality as two separate

inequalities and solve the problem.

2 7 11

2 7 7 11 7

2 4

2

x

x

x

x

or

2 7 11

2 7 7 11 7

2 18

9

x

x

x

x


Interval Notation: ( , 2) (9, )

x x x

Example 16: Solve the absolute inequality 2 10 0x

Solution: In these types of problems when you see that the inequality expression is greater than zero,

remove the absolute symbol, replace the inequality symbol with an equal sign and solve the equation.

2 10 0

2 10 0

2 10

5

x

x

x

x

The solution is all real numbers except 5 or except 5.

Interval notation: (- ,5) (5, ).

Now that we have discussed all types of absolute equalities and inequalities, we should be able to solve an

example with absolute equality and its related inequalities.

Example 17: Solve the absolute equality and its given related inequalities.

a) 2 3 7 b) 2 3 7 c) 2 3 7 x x x

Solution:

a) 2 3 7

2 3 7 or 2 3 7

2 7 3 or 2 7 3

2 4 or 2 10

2 or 5

{ 2,5}

x

x x

x x

x x

x x

x

b) 2 3 7

7 2 3 7

7 3 2 7 3

4 2 10

2 5

Interval Notation: [ 2,5]


x

x

x

x

x

x x

c) 2 3 7

2 3 7 or 2 3 7

2 7 3 or 2 7 3

2 4 or 2 10

2 or 5

Interval Notation: ( , 2] [5, )


x

x x

x x

x x

x x

x x x


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Example 18: Use your graphing calculator to solve the absolute equality and its related inequalities.

a) 1 6 10 b) 1 6 10 c) 1 6 10 x x x x x x

Solution: First enter the left side of the equality as Y1 and the right hand side as Y2.

Move cursor close to

the other intersection

From the final screen we can see that:

a) 2.5,7.5 b) [-2.5,7.5] c) ( , 2.5] [7.5, )x


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Exercises 2.3

In exercises 1 – 16, solve the absolute equations.

1) 5 4 2) 2 3 3) 2 3 5 4) 3 1 7x x x x

5) 2 7 2 8 6) 3 4 1 3 7) 5 2 6 10 8) 4 5 4 3x x x x

9) 3 2 5 2 10) 5 1 9 5 11) 2 6 5 5 12) 3 9 7 7x x x x

13) 3 2 5 14) 3 5 5 2 15) 2 1 2 3 16) 5 4 5 1x x x x x x x x

In exercises 17 – 40, solve the absolute inequalities.

17) 4 3 18) 1 8 19) 2 3 5 20) 4 3 1x x x x

21) 2 4 3 5 22) 3 2 4 6 23) 2 10 5 5 24) 2 6 3 3x x x x

25) 5 2 6 2 26) 6 5 7 2 27) 2 10 1 1 28) 2 6 4 4x x x x

29) 2 5 30) 3 4 31) 3 2 5 32) 5 4 1x x x x

33) 2 3 4 2 34) 3 5 2 8 35) 2 10 11 11 36) 2 6 7 7x x x x

37) 5 2 6 2 38) 6 5 7 2 39) 2 10 9 9 40) 2 6 5 5x x x x

In exercises 41 –46, solve the absolute equation and its related inequalities (see example #17).

41. a) 1 3 42. a) 2 6 43. a) 1 3 9

b) 1 3 b) 2 6 b) 1 3 9

x x x x x x

x x x x x x

c) 1 3 c) 2 6 c) 1 3 9x x x x x x

44. a) 4 3 5 45. a) 5 3 11 46. a) 4 2 5

b) 4 3 5 b) 5 3 11 b) 4 2 5

c) 4 3 5

x x x x x x

x x x x x x

x x

c) 5 3 11 c) 4 2 5x x x x


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2.4 Piecewise Functions

Definition: A piecewise function is a defined function that consists of several functions (i.e. The graph of

a piecewise function consists of several graphs. Each one of these graphs is also a function). Graphs of

parts of a piecewise function usually have one or more of the following: holes, sharp corners, and jumps.

Example 1: Graph the following piecewise function and evaluate it at the given values.

2 , if 0

; 3 , 2 , 02 5, if 0

x xf x f f f

x x

Solution: To graph the given piecewise function, we need to graph each part separately. Be sure to graph

each part within its defined domain. You can see that the first part which is a parabola is defined only

over the interval ( ,0] and the point zero is included (closed circle). The second part represents a line.

The line is defined only on the interval (0, ) where point zero is excluded (open circle).

Now, press TRACE and enter the values for x at which you want to evaluate the function f.

2

2

3 2 3 5 1, use the second function since 3 is in its domain only.

2 2 4, use the first function since -2 is in its domain only

0 0 0, use the first function since 0 is in its domain only

f

f

f

:Computer generated graph

Section 2.4 Piecewise Functions

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Note: There are other ways to enter the piecewise function of example #1 in our TI. See the following

screens to see how this can be done. In our opinion, the first method is easier than these other methods.

Method 2 Method 3

Greatest Integer Function (Step Function):

Notation: f x x ; which means the greatest integer that is less than or equal to x. Equality will occur

only when x is an integer. If not, always round down.

Example 2: Evaluate the following: 3.4 , 5 , 1.2 , 0 , 1.45 , 7f x

Solution: Since the value of the function is equal to x if x is an integer, we need to round down if x is not

an integer, 3, 5, 1,0 , 2 , 7f x

Graphing Calculator Solution of Example 2: Press MATH NUM 5:int(

Graph of the Greatest Integer Function f x x :

Domain: The set of all real numbers or ,

Range: The set of all integers

G G :Computer enerated raph


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Using a graphing calculator to graph f x x :

To find "int", press MATH NUM 5:int(

Don’t be deceived by the given TI graph. This is a lack of technology. All vertical segments are not part

of the graph. If you want to see a better graph, you need to change the MODE in your TI from Connected

to Dot. To do this, press MODE, move the cursor down to Dot and press ENTER to highlight Dot. If you

do this you’ll get the screen below. Now you can see why we call this function “Step Function”.

Note: The TI graphing calculator will not show the open or closed end of each segment.

Example 3: Graph the function 1 , evaluate 1.3 , 3 , 2.3 , 4 ,and 7.1f x x f f f f f

Solution: Since the value of the function is equal to x if x is an integer, round down if x is not an integer.

1.3 1.3 1 2.3 3

3 3 1 4 4

2.3 2.3 1 1.3 1

4 4 1 3 3

7.1 7.1 1 6.1 6

f

f

f

f

f


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Using the Table Feature:

Press 2nd TBLSET , highlight Ask to the right of Indpnt: , and highlight Auto to the

right of Depend: .

Press 2nd TABLE , enter the value for x at which you want to evaluate the function, and then

press ENTER .

From the last screen, 1.3 3, 3 4, 2.3 1, 4 3, and 7.1 6f f f f f .

G G :Computer enerated raph


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Exercises 2.4

In exercises 1 – 10, graph the function and evaluate it at the given values.

3 if 2

1) , evaluate 4 , 5 ,and 25 if 2

x xf x f f f

x x

2 3 if 1

2) , evaluate 4 , 5 ,and 11 4 if 1

x xf x f f f

x x

1 2 if 1

3) , evaluate 0 , 1 , 1 ,and 22 3 if 1

x xf x f f f f

x x

3 2 if 2

4) , evaluate 0 , 2 , 2 ,and 32 3 if 2

x xf x f f f f

x x

2 if 0

5) , evaluate 0 , 1 ,and 14 3 if 0

x xf x f f f

x x

2

2 3 if 06) , evaluate 0 , 1 ,and 1

if 0

x xf x f f f

x x

2 if 1

7) , evaluate 3 , 1 ,and 23 5 if 1

x xf x f f f

x x

2

2 3 if 28) , evaluate 0 , 2 ,and 3

if 2

x xf x f f f

x x

2

2 1 if 2

9) if 2 1 , evaluate 0 , 2 , 1 , 3 , and 4

3 1 if 1

x x

f x x x f f f f f

x x

2

5 if 1

10) if 1 2 , evaluate 0 , 1 , 2 , 3 ,and 4

9 if 2

x x

f x x x f f f f f

x x


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In exercises 11 – 18, graph the function and evaluate it at the given values.

11) 2 , evaluate 2.3 , 5 , 4.1 , 3 ,and 5.1f x x f f f f f


13) 1 , evaluate 1.2 , 5.3 , 0 , 2.5 ,and 7f x x f f f f f


15) 3 1 , evaluate 1.2 , 2 , 2.1 , 3 ,and 4.1f x x f f f f f

16) 3 1 , evaluate 4.2 , 1 , 0 , 5 ,and 3.1f x x f f f f f

17) 2 , evaluate 3.2 , 1 , 0 , 2 ,and 7.2f x x f f f f f

18) 3 , evaluate 5.2 , 2 , 0 , 2.3 ,and 5f x x f f f f f


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2.5 Algebra of Functions and their Composites

Two functions f and g can be combined to form a new function h exactly the same way we add,

subtract, multiply, and divide real numbers.

1

2

Let and be two functions with domains and respectively then:

( ) domain { | }

( )

f g A B

h x f g x f x g x x x A B

h x f g x f x g x

Algebra of Functions

3

4

domain { | }

( ) domain { | }

( ) domain { | and 0}

x x A B

h x fg x f x g x x x A B

f xfh x x x x A B g x

g g x

The graph of the function ( ) ( )( )h x f g x can be obtained from the graph of f and g by adding the

corresponding y-coordinates as shown on the figure below.

Similarly, we can find the difference, product, and division of f and g. In case of division, remember

that ( ) 0.g x So any value of x that makes the denominator equal to zero must be excluded.

Example 1: If ( ) 2 5 and ( ) 3, find ( )( ), ( )( ), ( )( ), the ( )( ).f

f x x g x x f g x f g x fg x xg

Also evaluate ( )(1) and ( )(2)f

f gg

.

Solution:

2

( ) ( ) ( ) 2 5 3 3 2

( ) ( ) ( ) 2 5 3 2 5 3 8

( ) ( ) ( ) 2 5 3 2 15

( ) 2 5( ) , 3

( ) 3

To evaluate (1), and (2) :

(1) 3(1) 2 1

f g x f x g x x x x

f g x f x g x x x x x x

fg x f x g x x x x x

f f x xx x

g g x x

ff g

g

f g

2(2) 5 1

(2)2 3 5

f

g

Section 2.5 Algebra of Functions and their Composites

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Composition of Functions

Notation: f g x f g x

The above can be read in many ways. It can be read as “f composed with g at x” , “f composed with g”,

“the composition of f and g”, “f of g of x”, or f circle g.

The Composite Function is defined by

Its domain is all numbers in the domain of

such that is in the domain of

f g

f g x f g x

x g x

g x f

Example 2: If 3 5 and 2 3, find and .f x x g x x f g x g f x

Solution: 3 5 3 2 3 5 6 4f g x f g x g x x x

2 3 2 3 5 3 6 7g f x g f x f x x x

Note: In general f g x g f x

Example 3: If 2 7 and 5 1, find 3f x x g x x f g

Solution: We can use two different methods to solve the problem.

First Method:

Find and then substitute 3 for

2 7 2 5 1 7 10 5

3 10(3) 5 25

f g x x

f g x f g x g x x x

f g

Second Method:

3 3 , evaluate 3 and substitute the result into

3 5 3 1 15 1 16

3 3 16 2 16 7 32 7 25

f g f g g f

g

f g f g f

x g x

Domain of

Range of

f

g

f g x

Range of fDomain of g

g f

f g


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The Difference Quotient

The difference quotient expression is very important in the study of calculus. Looking at the graph below,

the difference quotient is the slope of the line PQ which is called the secant line.

From the graph, we see that the slope of secant line PQ is change in

change in PQ

f x h f xy Risem

x Run h

.

f x h f x

mh

Difference Quotient

Example 4: 2Let 3 5, find the difference quotient and simplify your answerf x x x

Solution: For clarity, we will solve this example step by step.

2

2 2 2

2 2 2

2

( ) 3 5

3 5 2 3 3 5

2 3 3 5 3 5

f x x x

f x h x h x h x xh h x h

f x h f x x xh h x h x x

f x h f x x

22 3xh h x 3 5h 2x 3x 5

22 3 (2 3)

(2 3)2 3

f x h f x xh h h h x h

f x h f x h x hm x h

h h

Note 1: The simplified form of the numerator f x h f x should have h as a common factor.

Note 2: Review your work if all of the terms of the original functions f(x) don’t cancel in the final

expression of the difference quotient.

x

y

f x

Secant line

,Q x h f x h

,P x f x

h

f x h f x

x x h

Section 2.5 Algebra of Functions and their Composites

-80-

Exercises 2.5

Let 2 3f x x and 4 5g x x . Find the following.

1. 2. 3. 4. 5. 1

6. 2 7. 2 8. 0 9. 3 10.

ff g x f g x f g x x f g

g

ff g f g g f

g

1

g

f

Let 2 4f x x and 3g x x . Find the following.

11. 12. 13. 14. 15. 2

16. 1 17. 3 18. 0 19. 1 20. 4

ff g x f g x f g x x f g

g

f gf g f g g f

g f


21. 22. 23. 24.

25. 1 26. 2 27. 1

f g x g f x f f x g g x

f g g f f f 28. 0g g


29. 30. 31. 32.

33. 2 34. 0 35. 2

f g x g f x f f x g g x

f g g f f f 36. 1g g

In exercises 37 – 44, find the difference quotient of each given function and simplify the answer

completely.

37. 2 5 38. 3 4

39. 5 4

f x x f x x

f x x

2 2

2

40. 3 7

41. 3 1 42. 2 1

43. 2 3

f x x

f x x x f x x x

f x x x

2 44. 4 1f x x x


-81-

Answers - Exercises 2.1

1. Odd 2. Even 3. Even 4. Neither 5. Odd 6. Neither 7. Even

8. Odd 9. Neither 10. Even 11. Odd 12. Odd 13. y-axis 14. Origin

15. Origin 16. Origin 17. y-axis 18. All 19. All 20. All 21. All

22. All 23. x-axis 24. y-axis


1. 2

2 2g x x 2. 2

3 3g x x 3. 2 2 2g x x x 4. 2 4 3g x x x

5. 2 2 1g x x x 6. 2 4 1g x x x 7. 5 3 4g x x 8. 1 2 3g x x

9. 10. 11. 12.

13. 14. 15. 16.

17. 18. 19. 20.

21. 2

2 2 6g x x 22. 21 5

32 2

g x x

23. 2 1 4g x x

24. 4 4 12g x x

Answers to Chapter 2 Problems

-82-


1. 1,9x 2. 1,5x 3. 4,1x 4. 8

,23

x

5. 1 13

,2 2

x

6. 2

2,3

x

7. 6 2

,5 5

x

8. 9. 10. 11. 3x

12. 3x 13. 2

8,3

x

14. 3 7

,8 2

x

15. 1

2x

16. 3

10x

17. 1,7 18. 9,7 19. 1,4 20. 1

,12

21. 6,2 22. 8

,43

23. 24. 25. 26. 27. 5x 28. 3x

29. , 3 7, 30. , 7 1, 31. 7

, 1,3

32. 3

, 1,5

33. 9 3

, ,2 2

34. 5

, 5 ,3

35. ,5 5,x

36. ,3 3,x 37. 38. 39. 40.

41. a) 2,1 , b) , 2 1, , c) 2,1 42. a) 4,2 , b) , 4 2, , c) 4,2

43. a) 5.5,3.5 , b) , 5.5 3.5, , c) 5.5,3.5 44. a) , b) , c)

45. a) 6.5,4.5 , b) , 6.5 4.5, , c) 6.5,4.5 46. a) , b) , c)


1. 1, -8, 3 2. -15, -7, 1 3. 1, 3, 5, 8 4. 2, -4, 1, 3

5. 3, 1, 7 6. 0, -5, 1 7. 4, 1, 4 8. 0, -7, -9


-83-

9. 0, -3, 2, -5, 11 10. 0, -6, 7, -8, 5 11. -5, -7, 2, 1, 3 12. -3, -4, 1, 3, 6

13. -1, -5, 1, 3, 8 14. -2, -2, 1, 8, 10 15. 2, 2, 1, 7, 8 16. -9, -5, -4, 1, -1

17. -2, 1, 2, 4, 9 18. -9, -5, -3, -1, 2


1) 6 2x 2) 2 8x 3) 28 2 15x x 4)

2 3 5,

4 5 4

xx

x

5) 8 6) 13 7) 4 8) 8

9) 3

17 10)

1

5 11) 3 1x 12) 7x 13)

22 2 12x x 14) 2 4

; 33

xx

x

15) 7

16) 8 17) 4 18) 7 19) 3 20) 1

12 21) 8 11x 22) 8 1x 23) 16 5x

24) 4 9x 25) 19 26) 15 27) 21 28) 9 29) 29 6 2x x 30)

23 10x

31) 4 26 6x x 32) 9 4x 33) 22 34) 10 35) 2 36) 5 37) 2

38) 3 39) 5 40) 3 41) 2 3x h 42) 2 2x h 43) 2 2x h 44) 2 4x h

Chapter 2 Analysis of Graphs of Functions - Cerritos...

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