8/24/2024

Thomas’ Calculus: Early Transcendentals

Fifteenth Edition

Chapter 1
Functions

Slide - 1
 Section 1.1
Functions and Their Graphs

Examples Exercises
2, 4, 7, 8 3,5,6(only domains), 15, 18,55

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 Functions; Domain
Functions; and
Domain Range
and Range

Definition
𝑫 𝒀
A function 𝑓 from a set 𝐷 to a set 𝑌 is a 1
a
rule that assigns a unique value 𝑓(𝑥) in 𝑌 2
b
to each 𝑥 in 𝐷. 3
c 4
Domain is the value of 𝑥.

is the set of all possible values of 𝑓 𝑥

Range as 𝑥 varies throughout the domain.

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Functions; Domain and Range

A diagram showing a function as a kind of machine.

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Functions; Domain and Range

A function from a set 𝐷 to a set 𝑌 assigns a unique element

of 𝑌 to each element in 𝐷.

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 Graphs of Functions

Example 2

Graph the function 𝑦 = 𝑥 2 over the interval [−2 , 2] .

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 The Vertical Line Test for a Function

A curve in the 𝑥𝑦 plane is the graph of a function of 𝑥 if

no vertical line intersects the curve more than once.

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Piecewise-Defined Functions

Sometimes a function is described in pieces by using different

formulas on different parts of its domain. One example is the
absolute value function

𝑥 𝑖𝑓 𝑥 ≥ 0
𝑓(𝑥) = 𝑥 = ቐ
−𝑥 𝑖𝑓 𝑥 < 0

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Properties of Absolute Values
Suppose 𝑎 > 0 is any real number .Then

1 𝑥 = −𝑥

2 𝑥 ≤ 𝑎 ⟺ −𝑎 ≤ 𝑥 ≤ 𝑎 . { 𝑥 ∈ −𝑎, 𝑎 }

3 𝑥 ≥ 𝑎 ⟺ 𝑥 ≥ 𝑎 𝑜𝑟 𝑥 ≤ −𝑎 . { 𝑥 ∈ −∞, −𝑎 ∪ 𝑎, ∞ }

4 𝑥 = 𝑎 ⟺ 𝑥 = ±𝑎

𝑎2 = 𝑎 if 𝑎 ≥ 0. 𝑎2 = −𝑎 if 𝑎 < 0.
.
𝑎2 = 𝑎
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Example 4
 − x, x  0
 2
Sketch the function f ( x) =  x , 0  x  1
1, x  1


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Increasing and Decreasing Functions

1 A function 𝑓 is called increasing on an interval 𝐼,

if 𝑓(𝑥1 ) < 𝑓(𝑥2 ) whenever 𝑥1 < 𝑥2 in 𝐼.

2 A function 𝑓 is called decreasing on an interval 𝐼,

𝑖𝑓 𝑓(𝑥1 ) > 𝑓(𝑥2 ) , 𝑥1 < 𝑥2 in 𝐼 .

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Example 7

Determine whether the following graph is increasing, decreasing

or neither.

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 24 August 2024 13

Example

The function whose graph is given is

(a) Increasing on −∞, 0 .

(b) Increasing on 0, ∞ .

(c) decreasing on −∞, 0 .

(d) decreasing on ℝ.

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 Even Functions and Odd Functions: Symmetry

Even Function Odd Function

If 𝑓 −𝑥 = 𝑓(𝑥) ∀ 𝑥 ∈ 𝐷, then If 𝑓 −𝑥 = −𝑓 𝑥 ∀ 𝑥 ∈ 𝐷, then 𝑓 is

𝑓 is called an even function. called an odd function.

The graph is symmetric with The graph is symmetric with respect

respect to the _ _ _ _ _ _ . to the _ _ _ _ _ _ .

𝑓 𝑥 = 𝑥 2 is even 𝑓 𝑥 = 𝑥 3 is odd

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 Special Properties

Note

even ± even even

odd ± odd odd
odd ± even neither

even × even even

÷
odd × odd even
÷
odd × even odd
÷

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 Determine whether each of the following
Example 8
functions is even, odd or neither even nor odd.

(a) 𝑓 𝑥 = 𝑥 2 (b) 𝑓 𝑥 = 𝑥 2 + 1 (c) 𝑓 𝑥 = 𝑥 (d) 𝑓 𝑥 = 𝑥 + 1

Exercise 55
Note
1
ℎ 𝑡 = (a) 𝑓 𝑥 = 𝑐 𝑖𝑠 _ _ _ _ _
𝑡−1
(b) 𝑓 𝑥 = 𝑥 is _ _ _ _ _

(c) 𝑓 𝑥 = 𝑥 𝑛 𝑖𝑠 ቊ −− − 𝑖𝑓 𝑛 is even
_ _ _ _ _, 𝑖𝑓 𝑛 is odd

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 ALGEBRAIC FUNCTIONS

Any function constructed from

Polynomial Functions
polynomials using algebraic
Rational Functions operations (such as +, −,×,÷,
𝒏
)

Radical Functions starting with polynomials.

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 Common Functions

Linear Functions

A function of the form 𝒇(𝒙) = 𝒎𝒙 + 𝒃, where 𝑚 and 𝑏 are fixed constant is

called a linear function. The function 𝑓(𝑥) = 𝑥 where 𝑚 = 1 and 𝑏 = 0 is
called the identity function. When 𝑚 = 0 , then 𝑓(𝑥) = 𝑏 is called the
constant function.

(a) Lines through the origin with (b) A constant function with
slope 𝑚. slope 𝑚 = 0
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 Power Function is a function of the from 𝑓 𝑥 = 𝑥 𝑎

a is a constant

we have 3 cases

1 𝑎 = 𝑛 ( +𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑒𝑥𝑎𝑚𝑝𝑙𝑒: 𝑥 1 , 𝑥 2 , 𝑥 5 , … )

𝑫 = ℝ = (−∞, ∞) 𝑫 = (−∞, ∞) 𝑫 = (−∞, ∞) 𝑫 = (−∞, ∞) 𝑫 = (−∞, ∞)

𝑹 = ℝ = (−∞, ∞) 𝑹 = [𝟎, ∞) 𝑹 = (−∞, ∞) 𝑹 = [𝟎, ∞) 𝑹 = (−∞, ∞)

𝒏 𝒊𝒔 𝒐𝒅𝒅 𝒏 𝒊𝒔 𝒆𝒗𝒆𝒏 𝒏 𝒊𝒔 𝒐𝒅𝒅 𝒏 𝒊𝒔 𝒆𝒗𝒆𝒏 𝒏 𝒊𝒔 𝒐𝒅𝒅

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 1 1
2 𝑎 = −𝑛 (𝑒𝑥𝑎𝑚𝑝𝑙𝑒: , 2 , … . )
𝑥 𝑥
reciprocal

𝑫 = (−∞, 𝟎) ∪ (𝟎, ∞) 𝑫 = (−∞, 𝟎) ∪ (𝟎, ∞)

𝑹 = (−∞, 𝟎) ∪ (𝟎, ∞) 𝑹 = (𝟎, ∞)
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 1 1 1

3 𝑎= ( +𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑒𝑥𝑎𝑚𝑝𝑙𝑒: 𝑥 , 𝑥 , … ) 2 3

𝑛
Root function

𝑫 = 𝟎, ∞ 𝑫 = ℝ = (−∞, ∞) 𝑫 = 𝟎, ∞ 𝑫 = (−∞, ∞)
𝑹 = 𝟎, ∞ 𝑹 = ℝ = (−∞, ∞) 𝑹 = 𝟎, ∞ 𝑹 = 𝟎, ∞

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 24 August 2024 22

Polynomial Function

A function 𝑓(𝑥) is called a polynomial if constant

f(x) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎1 𝑥 + 𝑎0

x is a variable
Leading coefficient coefficients

Examples
𝑛: non-negative integer
2 2
𝑎𝑛 , 𝑎𝑛−1 , … , 𝑎1 , 𝑎0 ∈ ℝ (1) 𝑓 𝑥 = 𝑥 5
+ 𝑥 + 1 is a polynomial
3
Domain of f(x) = ℝ = −∞, ∞ . deg.( 𝒇 𝒙 ) =
Degree of the polynomial = 𝑛.
(2) 𝑔 𝑥 = 𝑥 + 2𝑥 −1 + 𝑥 2∕3
is not a polynomial

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1 Constant Function 3 Quadratic Function

𝑓 𝑥 =𝑐 𝑓 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 Deg. 𝑓(𝑥) =

Deg. 𝑓(𝑥) =
𝐷𝑓 =
𝑅𝑎𝑛𝑔𝑒 =

𝑎>0 𝑎<0
2 Linear Function
4 Cubic Function
𝑓 𝑥 = 𝑚𝑥 + 𝑏
𝑓 𝑥 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑
Deg. 𝑓(𝑥) =
Deg. 𝑓 𝑥 =
Domain =
Range= Domain =
Range=

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Graphs of three polynomial functions.

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 25

Rational Function

A rational function is a ratio of two polynomial functions:

𝑝 𝑥 The Domain of the rational function :
𝑓 𝑥 = , 𝑞 𝑥 ≠0
𝑞 𝑥 𝐷𝑓 = 𝑥 ∈ ℝ: 𝑞 𝑥 ≠ 0 = ℝ − {𝑧𝑒𝑟𝑜𝑠 𝑜𝑓 𝑞 𝑥 }

Graphs of three rational functions. The straight red lines approached by the
graphs are called asymptotes and are not part of the graphs.

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 26

Examples

3𝑥 − 1
ℎ 𝑥 =
2𝑥 4 − 𝑥 2 + 1 3 𝑥 2 + 5𝑥 − 6
1 𝑓 𝑥 =
𝑥2 − 4

𝐷𝑓 = 𝐷𝑓 =

1
2 𝑔 𝑥 =
4
5𝑥 − 3 4 𝑘 𝑥 =
𝑥2 − 𝑥
𝑥2 + 4

𝐷𝑓 =
𝐷𝑓 =

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 19/02/1446 27

Radical Function
𝑛
A radical function is an algebraic function of the form : 𝑅(𝑥) = 𝑓 𝑥
where 𝑓(𝑥) is a polynomial.

𝒏 ℝ 𝒊𝒇 𝒏 = 𝒐𝒅𝒅
𝒅𝒐𝒎𝒂𝒊𝒏 𝒐𝒇 𝒇 𝒙 :ቊ
𝒙∈ℝ ∶𝒇 𝒙 ≥𝟎 𝒊𝒇 𝒏 = 𝒆𝒗𝒆𝒏

Example Find the domain of

3
𝑎 𝑓 𝑥 = 𝑥+2 𝑏 𝑓 𝑡 = 2𝑡 − 1

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 28

Important Functions

1 𝑓 𝑥 = 𝑎2 − 𝑥 2 2 𝑓 𝑥 = 𝑥 2 − 𝑎2 3 𝑓 𝑥 = 𝑥 2 + 𝑎2

(0,a)
-a a -a a

domain=−− − domain=−−−− − domain = −− −

range=−− − range=−− − range = −− −
example example example
ℎ 𝑥 = 4 − 𝑥2 𝑓 𝑥 = 𝑥2 + 5
𝑓 𝑥 = 𝑥2 −9
domain domain domain
Range Range Range
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 Algebraic Functions

Any function constructed from polynomials using algebraic operations

(addition, subtraction, multiplication, division, and taking roots) lies
within the class of algebraic functions. All rational functions are
algebraic, but also included are more complicated functions (such as
those satisfying an equation like 𝑦 3 − 9𝑥𝑦 + 𝑥 3 = 0.

The figure below displays the graphs of three algebraic functions.

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 Exercise 3 Find the domain of Exercise 6 Find the domain of

2
𝐹 𝑥 = 5𝑥 + 10 𝐺 𝑡 = 2
𝑡 − 16

Exercise 5 Find the domain of

4
𝑓 𝑡 =
3−𝑡

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 Exercise 15 Exercise 18

Find the natural domain and Find the natural domain and
graph the function graph the function

𝑓 𝑥 = 5 − 2𝑥 𝑓 𝑥 = −𝑥

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TRANSCENDENTAL FUNCTIONS

Functions that are not algebraic are called transcendental;

Trigonometric Functions

Exponential Functions

Logarithm Functions
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Trigonometric Functions

The graphs of the sine and cosine functions are shown below.

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Exponential Functions

are of the form of 𝑓 𝑥 = 𝑎 𝑥 , 𝑎 is a base > 0 , 𝑎 ≠ 1.

Domain=
Domain= Range=
Range=

𝟏 𝒙
𝒂>𝟏, 𝒚= 𝟐𝒙 , 𝒚 = 𝒆𝒙 𝟎 < 𝒂 < 𝟏, 𝒚= 𝟐

Increasing or decreasing !!
even or odd ?
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 ‫العدد‬
Logarithmic Functions 𝑓 𝑥 = 𝑙𝑜𝑔𝑎 𝑥

The power to which we The base 𝒂 is ‫الأساس‬

raise 𝒂 to get 𝒙 ‫القوى‬ a +ve constant ≠ 𝟏

Note

𝑦 = 𝑙𝑜𝑔𝑒 𝑥 = ln 𝑥
1
𝑙𝑜𝑔 functions are the inverse functions
2 of the exponential functions.
Domain= Range=
3 𝑙𝑜𝑔𝑎 1 = ⋯ , 𝑙𝑜𝑔𝑎 𝑎 = ⋯
Increasing or decreasing !!
even or odd ?
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 HOMEWORK

1,2, 17, 19, 49, 53

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