A function 𝑷 is called a polynomial if

(1)

Domain and Range of a Function:

We call the domain of a function 𝒇 is 𝑫_𝒇 and range of 𝒇 is 𝑹_𝒇 .

A Catalog of Essential Functions

1- Linear Functions:

It has the form

𝒚 = 𝒇(𝒙) = 𝒎𝒙 + 𝒃

𝑫_𝒇= ℝ = (−∞, ∞) , 𝑹_𝒇= ℝ = (−∞, ∞)

2- Polynomial Functions:

A function 𝑷 is called a polynomial if

𝒚 = 𝑷(𝒙) = 𝒂

_𝒏

𝒙

^𝒏

+ 𝒂

_𝒏−𝟏

𝒙

^𝒏−𝟏

+ 𝒂

_𝒏−𝟐

𝒙

^𝒏−𝟐

+ ⋯ + 𝒂

_𝟑

𝒙

^𝟑

+ 𝒂

_𝟐

𝒙

^𝟐

+ +𝒂

_𝟏

𝒙 + 𝒂

_𝟎

,

where 𝒏 is a nonnegative integer and the numbers 𝒂

_𝟎

, 𝒂

_𝟏

, 𝒂

_𝟐

, … , 𝒂

_𝒏

are constants called the coefficients of the polynomial. If the leading coefficient 𝒂

_𝒏

≠ 𝟎, then the degree of the polynomial is 𝒏. For example, 𝑷(𝒙) = 𝟐𝒙

^𝟔

− 𝒙

^𝟒

+ 𝟐

𝟓 𝒙

^𝟑

+ √𝟐 𝑷(𝒙) = 𝒙

^𝟓

+ 𝟐√𝒙

^𝟑

− 𝒙

𝑷(𝒙) = 𝟒𝒙

^𝟐

− 𝟕 𝒙 + 𝟗

𝑫_𝑷= ℝ = (−∞, ∞) , 𝑹_𝑷: depends on the higher degree of the polynomial

There are four special cases of the polynomials:

 Constant Function:

𝑷(𝒙) = 𝒂

_𝟎

𝑫_𝑷= ℝ = (−∞, ∞) , 𝑹_𝑷 = 𝒂_𝟎

 Linear Function:

𝑷(𝒙) = 𝒂

_𝟏

𝒙 + 𝒂

_𝟎

𝑫_𝑷= ℝ = (−∞, ∞) , 𝑹_𝑷= ℝ = (−∞, ∞)

 Quadratic Function:

𝑷(𝒙) = 𝒂

_𝟐

𝒙

^𝟐

+ +𝒂

_𝟏

𝒙 + 𝒂

_𝟎

𝑫_𝑷= ℝ = (−∞, ∞)

 Cubic Function:

𝑷(𝒙) = 𝒂

_𝟑

𝒙

^𝟑

+ 𝒂

_𝟐

𝒙

^𝟐

+ +𝒂

_𝟏

𝒙 + 𝒂

_𝟎

𝑫_𝑷= ℝ = (−∞, ∞)

Example: Find the domain of the following functions:

(𝒂) 𝑓(𝑥) = 𝑥²− 3𝑥 + 2 (𝒃) 𝑓(𝑥) = 3 − 𝑥³ (𝒄) 𝑓(𝑥) = 7

(𝒅) 𝑓(𝑥) = √3𝑥 − 11 (𝒆) 𝑓(𝑥) = −4

9

(2)

3- Power Functions:

𝒚 = 𝒇(𝒙) = 𝒙

^𝒂

, 𝒂 𝐢𝐬 𝐚 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭.

We consider several cases.

(𝒊) 𝒂 = 𝒏, where 𝒏 is a positive integer

𝑫_𝒇= ℝ = (−∞, ∞)

The graphs of 𝒇(𝒙) = 𝒙

^𝒏

for 𝒏 = 𝟏, 𝟐, 𝟑, 𝟒, and 𝟓 are shown in the Figure below.

(𝒊𝒊) 𝒂 = −𝟏, −𝟐

^−𝟏

=

^𝟏_𝒙

& 𝒇(𝒙) = 𝒙

^−𝟐

=

_𝒙^𝟏_𝟐

are shown in the Figure below.

𝑫_𝒇= 𝑫_𝒇= 𝑹_𝒇= 𝑹_𝒇=

(𝒊𝒊𝒊) 𝒂 = 𝟏/𝒏, where 𝒏 is a positive integer

^𝟏^𝟐

= √𝒙 & 𝒇(𝒙) = 𝒙

^𝟏^𝟑

= √𝒙

^𝟑

are shown in the Figure below.

𝑫_𝒇= 𝑫_𝒇= 𝑹_𝒇= 𝑹_𝒇=

(3)

Generalizations:

(𝟏) If 𝒇(𝒙) = √𝒙^𝒏 {𝒏 𝐢𝐬 𝐨𝐝𝐝, 𝑫_𝒇= ℝ , 𝑹_𝒇= ℝ 𝒏 𝐢𝐬 𝐞𝐯𝐞𝐧, 𝑫_𝒇= [𝟎, ∞) , 𝑹_𝒇= [𝟎, ∞) (𝟐) If 𝒇(𝒙) = √𝒈(𝒙)^𝒏 {𝒏 𝐢𝐬 𝐨𝐝𝐝, 𝑫_𝒇= 𝑫_𝒈

𝒏 𝐢𝐬 𝐞𝐯𝐞𝐧, 𝑫_𝒇 = 𝑫_𝒈 𝐬𝐮𝐜𝐡 𝐭𝐡𝐚𝐭 𝒈(𝒙) ≥ 𝟎

Example: Find the domain and range of the function 𝑓(𝑥) = √𝑥

Solution:

Example: Find the domain and range of the function 𝑓(𝑥) = √𝑥 + 2

Solution:

Example: Find the domain of the function 𝑓(𝑥) = √1 − 𝑥

Solution:

Example: Find the domain of the function

𝑓(𝑥) = √2𝑥 − 5

³ Solution:

Generalizations:

(𝟏) If 𝒇(𝒙) = √𝒂 − 𝒙^𝟐 , then 𝑫_𝒇= [−√𝒂, √𝒂] , & 𝑹𝒇= [𝟎, √𝒂]

(𝟐) If 𝒇(𝒙) = √𝒙^𝟐− 𝒂 , then 𝑫_𝒇= (−∞, −√𝒂] ∪ [√𝒂, ∞) , & 𝑹𝒇= [𝟎, ∞)

Example: Find the domain and range of the function 𝑓(𝑥) = √49 − 𝑥²

Solution:

Example: Find the domain and range of the function 𝑓(𝑥) = √𝑥²− 2

Solution:

Example: Find the domain and range of the function

𝑓(𝑥) = √𝑥²+ 3 Solution:

(4)

4- Rational Functions:

A rational function 𝒇 is a ratio of two polynomials:

𝒇(𝒙) = 𝑷(𝒙) 𝑸(𝒙) , where 𝑷 and 𝑸 are polynomials.

𝑫

_𝒇

= ℝ\{𝒙: 𝑸(𝒙) = 𝟎} = ℝ\{ماقملا رافصأ}

Generalization:

If 𝒇(𝒙) =

^𝑷(𝒙)

𝑸(𝒙)

, then 𝑫

_𝒇

= 𝑫

_𝑷

∩ 𝑫

_𝑸

\{𝒙: 𝑸(𝒙) = 𝟎} = 𝑫

_𝑷

∩ 𝑫

_𝑸

\{ماقملا رافصأ}

Example: Find the domain of each of the following functions:

(𝑎) 𝑓(𝑥) = 1 𝑥

(𝑏) 𝑔(𝑥) = 𝑥 + 3 𝑥

²

− 1

(𝑐) ℎ(𝑥) = 1 𝑥

²

− 𝑥

(𝑑) 𝐹(𝑥) = 7 𝑥

²

+ 2

(𝑒) 𝐺(𝑥) = 𝑥 + 3 𝑥 + 5

(𝑓) 𝐻(𝑥) = 𝑥 + 1

𝑥

²

− 3𝑥 + 2

(5)

(𝑔) 𝑘(𝑥) = 2 − 𝑥 10 − 3𝑥 − 𝑥

²

5- Piecewise Defined Functions:

Generalization:

If 𝒇(𝒙) = |𝒈(𝒙)| , then 𝑫

_𝒇

= 𝑫

_𝒈

.

Example: A function

𝒇

is defined by

𝒇(𝒙) = {𝟏 − 𝒙, 𝐢𝐟 𝒙 ≤ −𝟏 𝒙^𝟐, 𝐢𝐟 𝒙 > −𝟏 Evaluate 𝒇(−𝟐), 𝒇(−𝟏), and 𝒇(𝟎) and sketch the graph.

Solution:

Example: Sketch the graph of the absolute value function

𝒇(𝒙) = |𝒙|

and find its domain and range.

Solution:

𝑫_𝒇= & 𝑹_𝒇= 6- The Greatest Integer Function:

(6)

Definition:

𝒇(𝒙) = ⟦𝒙⟧ 𝐢𝐬 𝐭𝐡𝐞 𝐥𝐚𝐫𝐠𝐞𝐬𝐭 𝐢𝐧𝐭𝐞𝐠𝐞𝐫 𝐧𝐮𝐦𝐛𝐞𝐫 𝐭𝐡𝐚𝐭 𝐢𝐬 𝐥𝐞𝐬𝐬 𝐭𝐡𝐚𝐧 𝐨𝐫 𝐞𝐪𝐮𝐚𝐥 𝐭𝐨 𝒙.

𝐈𝐟

𝒇

(

𝒙

)

=

⟦

𝒙

⟧, 𝐭𝐡𝐞𝐧 𝑫_𝒇= ℝ & 𝑹_𝒇= ℤ Example: Find

⟦𝟒⟧ =

⟦𝟎⟧ =

⟦𝟑. 𝟕⟧ =

⟦−𝟔⟧ =

⟦−𝟐. 𝟑⟧ =

Example: Find the domain and range of the function

(𝟏) 𝑓(𝑥) = ⟦𝑥⟧ 𝑫_𝒇= & 𝑹_𝒇= (𝟐) 𝑓(𝑥) = ⟦2𝑥 − 1⟧ 𝑫_𝒇= & 𝑹_𝒇= (𝟑) 𝑓(𝑥) = ⟦𝑥²− 2𝑥 + 3⟧ 𝑫_𝒇= & 𝑹_𝒇=

7- Algebraic Functions:

A function 𝒇 is called an algebraic function if it can be constructed using algebraic operations (such as addition, subtraction, multiplication, division, and taking roots) starting with polynomials. Any rational function is automatically an algebraic function. For example,

(𝟏) 𝑓(𝑥) = √𝑥²+ 1 (𝟐) 𝑔(𝑥) =𝑥⁴− 16𝑥²

𝑥 + √𝑥 + (𝑥 − 2)√𝑥 + 1³ (𝟑) 𝑓(𝑥) = |𝑥|

𝑥 − 9

(7)

(𝑎) 𝑓(𝑥) = √4 − 𝑥

²

𝑥 − 1

(𝑏) 𝑔(𝑥) = 1

√2 − 𝑥 + 5

(𝑐) ℎ(𝑥) = √ 𝑥 − 1 𝑥 + 1

(𝑑) 𝐹(𝑥) = ⟦𝑥

²

+ 2⟧

𝑥 − 3

8- Trigonometric Functions:

(𝟏) 𝒇(𝒙) = 𝐬𝐢𝐧 𝒙 𝑫_𝒇= ℝ 𝑹_𝒇= [−𝟏, 𝟏]

(𝟐) 𝒇(𝒙) = 𝐜𝐨𝐬 𝒙 𝑫_𝒇= ℝ 𝑹_𝒇= [−𝟏, 𝟏]

(𝟑) 𝒇(𝒙) = 𝐭𝐚𝐧 𝒙 𝑫_𝒇= ℝ\ {±𝝅 𝟐, ±𝟑𝝅

𝟐 , ±𝟓𝝅

𝟐 , … } 𝑹_𝒇= ℝ

(𝟒) 𝒇(𝒙) = 𝐜𝐨𝐭 𝒙 𝑫_𝒇= ℝ\{𝟎, ±𝝅, ±𝟐𝝅, ±𝟑𝝅, … } 𝑹_𝒇= ℝ

(𝟓) 𝒇(𝒙) = 𝐬𝐞𝐜 𝒙

𝑫_𝒇= ℝ\ {±𝝅 𝟐, ±𝟑𝝅

𝟐 , ±𝟓𝝅

𝟐 , … } 𝑹_𝒇= ℝ\(−𝟏, 𝟏)

(𝟔) 𝒇(𝒙) = 𝐜𝐬𝐜 𝒙 𝑫_𝒇= ℝ\{𝟎, ±𝝅, ±𝟐𝝅, ±𝟑𝝅, … } 𝑹_𝒇= ℝ\(−𝟏, 𝟏)

(8)

Graphs of the Trigonometric Functions

Remarks:

1- −𝟏 ≤ 𝐬𝐢𝐧 𝒙 ≤ 𝟏 2- −𝟏 ≤ 𝐜𝐨𝐬 𝒙 ≤ 𝟏

(9)

9- Exponential Functions:

The exponential functions are the functions of the form 𝒇(𝒙) = 𝒂

^𝒙

, where the base 𝒂 > 𝟎 & 𝒂 ≠ 𝟏.

𝐈𝐟

𝒇

(

𝒙

)

= 𝒂

^𝒙, 𝐭𝐡𝐞𝐧 𝑫_𝒇= ℝ & 𝑹_𝒇= (𝟎, ∞)

10- Logarithmic Functions:

The logarithmic functions are the functions of the form 𝒇(𝒙) = 𝐥𝐨𝐠

_𝒂

𝒙, where the base 𝒂 > 𝟎 & 𝒂 ≠ 𝟏. It is the inverse functions of the exponential functions.

𝐈𝐟

𝒇

(

𝒙

)

= 𝐥𝐨𝐠

_𝒂

𝒙

, 𝐭𝐡𝐞𝐧 𝑫_𝒇= (𝟎, ∞) & 𝑹_𝒇= ℝ

Example:

Classify the following functions as one of the types of functions that we have discussed.

(𝒂) 𝑓(𝑥) = 5^𝑥 (𝒃) 𝑔(𝑥) = 𝑥⁵ (𝒄) ℎ(𝑥) =

1 + 𝑥

1 − √𝑥

(𝒅) 𝑢(𝑡) = 1 − 𝑡 + 6𝑡²

(10)

Symmetry (Even & Odd Functions)

Definition:

The function 𝒚 = 𝒇(𝒙) is called

 An even function if 𝒇(−𝒙) = 𝒇(𝒙), ∀𝒙 ∈ 𝑫_𝒇

 An odd function if 𝒇(−𝒙) = −𝒇(𝒙), ∀𝒙 ∈ 𝑫_𝒇

Properties:

The function 𝒚 = 𝒇(𝒙) is an even function ⇔ its graph is symmetric about the 𝒚-axis.

The function 𝒚 = 𝒇(𝒙) is an odd function ⇔ its graph is symmetric about the origin.

Example:

Determine whether each of the following functions is even, odd, or neither even nor odd.

(𝒂) 𝑓(𝑥) = 𝑥⁴

− 𝑥

² Solution:

(𝒃) 𝑔(𝑥) = 𝑥⁵

+ 𝑥

³

− 6𝑥

Solution:

(𝒄) ℎ(𝑥) = 𝑥⁷

+ 11𝑥 − 9

Solution:

(𝒅) 𝑘(𝑥) = 2𝑥 − 𝑥² Solution:

(𝒆) 𝑟(𝑥) =

𝑥

²

− 1 𝑥

³

+ 4

Solution:

(𝒇) 𝑠(𝑥) =

3

𝑥

²

− 𝑥

Solution:

(11)

Increasing and Decreasing Functions

The graph shown in the Figure rises from 𝑨 to 𝑩, falls from 𝑩 to 𝑪, and rises again from 𝑪 to 𝑫.

The function is said to be increasing on the interval [𝒂, 𝒃], decreasing on [𝒃, 𝒄], and increasing again on [𝒄, 𝒅].

Definition:

A function

^𝒇

is called increasing on an interval 𝑰 if

𝒇(𝒙_𝟏) < 𝒇(𝒙_𝟐) 𝐰𝐡𝐞𝐧𝐞𝐯𝐞𝐫 𝒙_𝟏< 𝒙_𝟐 𝐢𝐧 𝑰.

It is called decreasing on 𝑰 if

𝒇(𝒙_𝟏) > 𝒇(𝒙_𝟐) 𝐰𝐡𝐞𝐧𝐞𝐯𝐞𝐫 𝒙_𝟏< 𝒙_𝟐 𝐢𝐧 𝑰.

Example:

𝑦 = 𝑥

²

𝑦 = 𝑥

³

(12)

𝑦 = √𝑥

𝑦 = 2

^𝑥

𝑦 = 2

^−𝑥

= ( 1 2 )

𝑥

𝑦 = |𝑥|

𝑦 = log

_𝑎

𝑥

(13)

Sections 1.1 & 1.2. Exercises Page 19-22 & 33

Homework: Page 19-21

1. If 𝑓(𝑥) = 𝑥 + √2 − 𝑥 and 𝑔(𝑢) = 𝑢 + √2 − 𝑢 , is it true that 𝑓 = 𝑔?

Determine whether the curve is the graph of a function of

𝒙

. If it is, state the domain and range of the function.

7. 8.

Find the domain of the function.

𝟑𝟏.

𝑓

⁽

𝑥

⁾

= 𝑥 + 4 𝑥

²

− 9

𝟑𝟐.

𝑓

(

𝑥

)

= 2𝑥

³

− 5

𝑥

²

+ 𝑥 − 6

𝟑𝟑.

𝑓

(

𝑡

)

=

³^√

2𝑡 − 1

𝟑𝟒.

𝑔

(

𝑡

)

=

^√

3 − 𝑡 −

√

2 + 𝑡

𝟑𝟖. Find the domain and range and sketch the graph of the function

ℎ(𝑥) = √4 − 𝑥

². Find the domain and sketch the graph of the function.

𝟒𝟓.

𝐺

(

𝑥

)

= 3𝑥 +

|

𝑥

|

𝑥

𝟒𝟕.

𝑓

(

𝑥

)

=

{

𝑥 + 2, 𝑥 < 0 1 − 𝑥, 𝑥 ≥ 0

Homework: Page 33

Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.

𝟏. (𝐚)

𝑓

(

𝑥

)

= log

₂

𝑥

(

𝐛

)

𝑔

(

𝑥

)

=

⁴√

𝑥

(𝐜)

ℎ

(

𝑥

)

= 2𝑥

³

1 − 𝑥

²

(

𝐝

)

𝑢

(

𝑡

)

= 1 − 1.1 𝑡 + 2.54 𝑡

² (𝐞)

𝑣

(

𝑡

)

= 5

^t

(

𝐟

)

𝑤

(

𝜃

)

= sin 𝜃 cos

²

𝜃

(14)

Tutorials: Page 19-22

3. The graph of a function

𝑓

is given.

(a) State the value of

𝑓(1)

. (b) Estimate the value of

𝑓(−1)

. (c) For what values of

𝒙

is

𝑓(𝑥) = 1

?

(d) Estimate the value of

𝒙

such that

𝑓(𝑥) = 0

. (e) State the domain and range of

𝑓

.

(f) On what interval is

𝑓

increasing?

Determine whether the curve is the graph of a function of

𝒙

. If it is, state the domain and range of the function.

9. 10.

Find the domain of the function.

𝟑𝟓.

ℎ

(

𝑥

)

= 1

√

𝑥

²

− 5𝑥

4

𝟑𝟕.

𝐹

(

𝑝

)

=

√

2 −

√

𝑝

Find the domain of the function. Determine whether

𝑓

is even, odd, or neither.

𝟕𝟑.

𝑓

(

𝑥

)

= 𝑥 𝑥

²

+ 1

𝟕𝟖.

𝑓

(

𝑥

)

= 1 + 3𝑥

³

− 𝑥

⁵ Tutorials: Page 33

Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.

𝟐. (𝐚)

𝑦 = 𝜋

^𝑥

⁽

𝐛

⁾

𝑦 = 𝑥

^𝜋

(𝐜)

𝑦 = 𝑥

²(

2 − 𝑥

³)

(

𝐝

)

𝑦 = tan 𝑡 − cos 𝑡

(𝐞)

𝑦 = s

1 + 𝑠

(

𝐟

)

𝑦 =

^√

𝑥

³

− 1

1 +

√³

𝑥