MATH 110
GENERAL EXAMPLES
Example: Find the domain of the given graph of the function
𝑦
420
5
-2-
Example: Find the range of the given graph of the function
1
𝑥
Example: If 𝑓(𝑥 ) = 2𝑥 2 − 4 , compute 𝑓(𝑎 + 1)
2
Example: Determine whether the below curves are the graph of
a function of x
By the vertical line test:
A curve in xy-plane is a graph of a function of x if and only if no
vertical line intersects the curve more than once.
3
Even and Odd functions:
Let 𝑓(𝑥 ) be a defined function, then
 𝑓 (𝑥 ) is an even function if
𝑓(−𝑥 ) = 𝑓(𝑥 ) , ∀ 𝑥 ∈ 𝐷𝑓
 𝑓 (𝑥 ) is an odd function if
𝑓(−𝑥 ) = −𝑓 (𝑥 ) , ∀ 𝑥 ∈ 𝐷𝑓
Useful Rules:
𝑜𝑑𝑑
1) 𝑒𝑣𝑒𝑛 = 𝑜𝑑𝑑
𝑒𝑣𝑒𝑛
3) 𝑒𝑣𝑒𝑛 = 𝑒𝑣𝑒𝑛
5) (−1)0𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 = −1
4
2)
𝑒𝑣𝑒𝑛
𝑜𝑑𝑑
= 𝑜𝑑𝑑
𝑜𝑑𝑑
4) 𝑜𝑑𝑑 = 𝑒𝑣𝑒𝑛
6) (−1)𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 = 1
Example: Determine whether 𝑓 (𝑥 ) is even, odd, or neither even
nor. odd functions.
1) 𝑓(𝑥 ) = 𝑥 5 + 2𝑥
2) 𝑓(𝑥 ) = 4𝑥 2 − 𝑥 4 + 4
𝑥 2 +1
3) 𝑓(𝑥 ) = 𝑥 7 −3𝑥 2
𝑥 3 −4𝑥
4) 𝑓(𝑥 ) = 𝑥 5 +2𝑥 9
5
Type of functions:
Polynomials:
General formula:
𝑓(𝑥 ) = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0 Where
𝑛 is a nonnegative integer
𝑎0 , 𝑎1 , 𝑎2 , … 𝑎𝑛 ∈ R (are constants called coefficients)
The degree of the polynomial is 𝑛
Domain of polynomials: is 𝑅 = (−∞, ∞)
Examples of polynomials:
1
𝑓(𝑥 ) = 4𝑥 3 − 2𝑥 2 + 2 𝑥 − √2
5
𝑓(𝑥 ) = √3𝑥 2 + 3 𝑥 + 1
Examples of not polynomials:
𝑓 (𝑥 ) = 𝑥 −2 + 4𝑥 − 1
1
2
𝑓 (𝑥 ) = 𝑥 + 4𝑥 + √3
6
Example: Specify the degree of the polynomial
𝑓(𝑥 ) = 4𝑥 2 + 7𝑥 5 − 1
Four types of polynomials:
7
1) 𝑓(𝑥 ) = 𝑎0
⟹ Constant function
2) 𝑓(𝑥 ) = 𝑎1 𝑥 + 𝑎0
⟹ Linear function
3) 𝑓(𝑥 ) = 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0
⟹ Quadratic function
4) 𝑓(𝑥 ) = 𝑎3 𝑥 3 + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0
⟹ Cubic function
Example: State the name of polynomials for the following
functions and specify its degree.
1
1) 𝑓(𝑥 ) = 2
3
2) 𝑓(𝑥 ) = 2 𝑥 2 + 𝑥 + 3
3) 𝑓(𝑥 ) = 4𝑥 2 + 1
4) 𝑓(𝑥 ) = 𝑥 3 − 3𝑥 2
8
5) 𝑓(𝑥 ) = 5𝑥 − 4
6) 𝑓(𝑥 ) = 3𝑥 − √3𝑥 2
9
Power function:
General formula:
𝑓 (𝑥 ) = 𝑥 𝑎 , 𝑤ℎ𝑒𝑟𝑒 𝑎 ∈ 𝑅
Examples:
𝑓(𝑥 ) = 𝑥
3
, 𝑓 (𝑥 ) = 𝑥
𝑓(𝑥 ) = 𝑥 1.3
1
2
, 𝑓 (𝑥 ) = 𝑥 −2
, 𝑓(𝑥 ) = 𝑥 √3
Root (radical) Functions:
General formula:
𝑓 (𝑥 ) = 𝑛√𝑃(𝑥)
, 𝑤ℎ𝑒𝑟𝑒 𝑃(𝑥 ) 𝑖𝑠 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙
and 𝑛 ≥ 2
10
Example: Determine the root functions from the following
functions
3
1) 𝑓(𝑥 ) = √2𝑥 2 + 4𝑥 − 1
2) 𝑓(𝑥 ) = √𝑥 −3 + 4𝑥
1
4
3) 𝑓(𝑥 ) = √𝑥 2 + 𝑥 5 − 2 𝑥 + 3
4
4) 𝑓(𝑥 ) = √√2𝑥 3 + 3 𝑥
6
1
5) 𝑓(𝑥 ) = √4𝑥 2 − 𝑥 3
1
6) 𝑓(𝑥 ) = √6𝑥 2 − 4𝑥 + 3
11
Domain of the Root functions:
𝑛
𝑓(𝑥 ) = √𝑃(𝑥) =
𝑅 = (−∞, ∞)
𝑖𝑓 𝒏 𝒊𝒔 𝒐𝒅𝒅
𝑃(𝑥 ) ≥ 0
𝑖𝑓 𝒏 𝒊𝒔 𝒆𝒗𝒆𝒏
Example: Find the domain of the below functions:
1) 𝑓(𝑥 ) = √𝑥 − 1
3
2) 𝑓(𝑥 ) = √6𝑥 2 − 1
5
3) 𝑓(𝑥 ) = √𝑥 − 1
12
4) 𝑓(𝑥 ) = √𝑥 2 − 4
13
5) 𝑓(𝑥 ) = √𝑥 2 − 5𝑥 + 6
14
Rational Function:
General formula:
𝑃(𝑥)
𝑓(𝑥 ) = 𝑄(𝑥) , where 𝑃(𝑥 )𝑎𝑛𝑑 𝑄 (𝑥 ) are polynomials
𝑄 (𝑥 ) ≠ 0.
Example: Determine the rational functions from the below
functions
1) 𝑓(𝑥 ) =
2) 𝑓(𝑥 ) =
3) 𝑓(𝑥 ) =
4) 𝑓(𝑥 ) =
5) 𝑓(𝑥 ) =
15
𝑥 2 −7𝑥 −1 +4
3𝑥 2 −4𝑥
1 2
𝑥 −4𝑥
2
2𝑥 3 −4𝑥 2 +1
√3𝑥 3 −4𝑥−7
1
2𝑥 2 +4𝑥 3
5𝑥 4 +3𝑥 2 −1
1
4
7𝑥 3 − 𝑥
2√𝑥+4𝑥 2
3𝑥 4 −2𝑥
⟹
⟹
⟹
⟹
⟹
Domain of Rational Functions:
𝐷 = 𝑅 − {𝑧𝑒𝑟𝑜𝑠 ‫}ماعدا اصفار المقام‬
Example: Find the domain of the below functions:
1) 𝑓(𝑥 ) =
2) 𝑓(𝑥 ) =
16
2𝑥 2 +7𝑥−4
𝑥−1
7𝑥 2 −2𝑥 3 +1
𝑥 2 −1
3) 𝑓(𝑥 ) =
17
5𝑥 2 −4𝑥+3
𝑥 2 +𝑥−6
Algebraic Functions:
All previous functions are algebraic functions.
Example: Find the domain of the below functions
1
1) 𝑓(𝑥 ) = 𝑥−3 + √𝑥 − 1
18
2) 𝑓(𝑥 ) =
19
√𝑥−4
√7−𝑥
Trigonometric Functions:
Examples:
𝑓(𝑥 ) = sin(2𝑥 − 1)
𝑓 (𝑥 ) = cos(3𝑥 2 − 1)
𝑓(𝑥 ) = sec(2𝑥 3 + 4)
𝑓(𝑥 ) = csc(4𝑥)
𝑓(𝑥 ) = cot(4𝑥 − 1)
𝑓(𝑥 ) = tan(2𝑥 − 1)
Exponential Functions:
General formula:
𝑓 (𝑥 ) = 𝑎 𝑥 , 𝑤ℎ𝑒𝑟𝑒 𝑎 ≠ 1 𝑎𝑛𝑑 𝑎 > 0
Examples:
𝑓(𝑥 ) = 4
𝑥
;
1 𝑥
𝑓(𝑥 ) = (2)
Logarithmic Functions:
General formula:
𝑓 (𝑥 ) = log 𝑎 𝑥
, 𝑤ℎ𝑒𝑟𝑒 𝑎 ≠ 1 𝑎𝑛𝑑 𝑎 > 0
Examples:
𝑓(𝑥 ) = log 3 𝑥
20
;
𝑓 (𝑥 ) = log 4 (2𝑥 2 − 1)
Transformation of functions:
Vertical and horizontal shifts:
Assume that ℎ > 0. If 𝑦 = 𝑓(𝑥) is the graph of a given function,
then:
1) The graph of 𝑦 = 𝑓 (𝑥 + ℎ) is shifted to the left h units.
2) The graph of 𝑦 = 𝑓 (𝑥 − ℎ) is shifted to the right h units.
3) The graph of 𝑦 = 𝑓 (𝑥 ) + ℎ is shifted upward h units.
4) The graph of 𝑦 = 𝑓 (𝑥 ) − ℎ is shifted downward h units.
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Example: Suppose that 𝑦 = 𝑓(𝑥) is a given function. Describe
the new graph of the below function in words.
1) 𝑦 = 𝑓(𝑥 ) − 3 ⟺ the graph is shifted downward 3 units.
2) 𝑦 = 𝑓(𝑥 ) + 3 ⟺ the graph is shifted upward 3 units.
3) 𝑦 = 𝑓(𝑥 − 3) ⟺ the graph is shifted to the right 3 units
4) 𝑦 = 𝑓(𝑥 + 3) ⟺ the graph is shifted to the left 3 units
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5) 𝑦 = 𝑓(𝑥 + 6) + 3 ⟺ the graph is shifted to the left 6 units
and then shifted upward 3 units.
6) 𝑦 = 𝑓(𝑥 − 6) + 3 ⟺ the graph is shifted to the right 6 units
and then shifted upward 3 units.
7) 𝑦 = 𝑓(𝑥 − 6) − 3 ⟺ the graph is shifted to the right 6 units
and then shifted downward 3 units.
8) 𝑦 = 𝑓(𝑥 + 6) − 3 ⟺ the graph is shifted to the left 6 units
and then shifted downward 3 units.
23
Reflecting Transformation:
If 𝑦 = 𝑓(𝑥) is the graph of a given function, then
1) The graph of 𝑦 = −𝑓(𝑥) is reflected the graph about x-axis
2) The graph of 𝑦 = 𝑓(−𝑥) is reflected the graph about y-axis
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Example: Suppose that 𝑦 = 𝑓(𝑥) is a given graph of the
function. Describe in words the following new functions
1) 𝑦 = 𝑓(−𝑥 ) ⟺ the graph is reflected about y-axis
2) 𝑦 = −𝑓 (𝑥 ) ⟺ the graph is reflected about x-axis
3) 𝑦 = −𝑓 (𝑥 ) + 6 ⟺ the graph is reflected about x-axis and
then shifted upward 6 units.
4) 𝑦 = 𝑓(−𝑥 ) − 6 ⟺ the graph is reflected about y-axis and
then shifted downward 6 units.
25
Arithmetic Combination of functions:
Suppose that 𝑓(𝑥 ) 𝑎𝑛𝑑 𝑔(𝑥) are functions, then
1) (𝑓 + 𝑔)(𝑥 ) = 𝑓(𝑥 ) + 𝑔(𝑥)
2) (𝑓 − 𝑔)(𝑥 ) = 𝑓(𝑥 ) − 𝑔(𝑥)
3) (𝑓. 𝑔)(𝑥 ) = 𝑓(𝑥 ). 𝑔(𝑥)
𝑓
𝑓(𝑥)
4) (𝑔) (𝑥 ) = 𝑔(𝑥)
,
𝑔(𝑥) ≠ 0
Example: If 𝑓(𝑥 ) = 𝑥 2 − 1 and 𝑔(𝑥 ) = 𝑥 − 1 , find:
1) (𝑓 + 𝑔)(𝑥 ) = 𝑓(𝑥 ) + 𝑔(𝑥)
2) (𝑓 − 𝑔)(𝑥 ) = 𝑓(𝑥 ) − 𝑔(𝑥)
26
3) (𝑓. 𝑔)(𝑥 ) = 𝑓 (𝑥 ). 𝑔(𝑥)
𝑓
𝑓 (𝑥 )
𝑔
𝑔 (𝑥 )
4) (𝑔) (𝑥 ) = 𝑔(𝑥)
5) (𝑓 ) (𝑥 ) = 𝑓(𝑥)
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Composition of functions:
Suppose that 𝑓(𝑥 ) 𝑎𝑛𝑑 𝑔(𝑥 ) are given functions. Then, the
composite functions is defined as:
(𝑓𝜊𝑔)(𝑥) = 𝑓(𝑔(𝑥 ))
Example: If 𝑓(𝑥 ) = 𝑥 2 + 1 and 𝑔(𝑥 ) = 𝑥 − 1 , find:
1) (𝑓𝜊𝑔)(𝑥) = 𝑓(𝑔(𝑥 )) = 𝑓(𝑥 − 1)
2) (𝑔𝜊𝑓)(𝑥) = 𝑔(𝑓(𝑥 )) = 𝑔(𝑥 2 + 1)
28
3) (𝑓𝜊𝑓)(𝑥) = 𝑓(𝑓(𝑥 )) = 𝑓(𝑥 2 + 1)
4) (𝑔𝜊𝑔)(𝑥) = 𝑔(𝑔(𝑥 )) = 𝑔(𝑥 − 1)
29
Domain of Composite functions:
𝐷{(𝑓𝜊𝑔)(𝑥)} =𝐷(𝑓𝜊𝑔)(𝑥) ∩ 𝐷𝑔(𝑥)
𝐷{(𝑔𝜊𝑓)(𝑥)} =𝐷(𝑔𝜊𝑓)(𝑥) ∩ 𝐷𝑓(𝑥)
30
and
Example: If 𝑓(𝑥 ) = √𝑥 − 4 and 𝑔(𝑥 ) = 𝑥 2 , find domain of
(𝑓𝜊𝑔)(𝑥)
( 𝐷{(𝑓𝜊𝑔)(𝑥)} )
𝐷{(𝑓𝜊𝑔)(𝑥)} =𝐷(𝑓𝜊𝑔)(𝑥) ∩ 𝐷𝑔(𝑥)
31
Exponential Function:
General formula:
𝑓 (𝑥 ) = 𝑎 𝑥 , 𝑤ℎ𝑒𝑟𝑒 𝑎 ≠ 1 𝑎𝑛𝑑 𝑎 > 0
Example: Determine the exponential functions of the following
functions:
a) 𝑓(𝑥 ) = 2𝑥
2 𝑥
b) 𝑓(𝑥 ) = (3)
c) 𝑓(𝑥 ) = (3.2)𝑥
d) 𝑓(𝑥 ) = (√3)
𝑥
1 𝑥
e) 𝑓(𝑥 ) = (− 2)
e) 𝑓(𝑥 ) = (1)𝑥
1 𝑥
f) 𝑓(𝑥 ) = − (2)
32
Domain of the exponential function:
Domain is 𝑅 = (−∞, ∞)
Example: Find the domain of the below functions
a) 𝑓(𝑥 ) = 3𝑥
4
b) 𝑓(𝑥 ) = ( )𝑥 − 5
7
c) 𝑓(𝑥 ) = −(√3)𝑥 + 4
The domain for any exponential function is always R
33
The range of the exponential functions:
Let 𝑎𝜖𝑅 and 𝑏 > 0 and 𝑏 ≠ 1. Then:
If 𝑦 = 𝑏 𝑥 ± 𝑎
⟹ 𝑅{𝑦} = (±𝑎, ∞)
If 𝑦 = −𝑏 𝑥 ± 𝑎 ⟹ 𝑅{𝑦} = (− ∞, ±𝑎)
Example: Find the range of the below functions:
1) 𝑓(𝑥 ) = 2𝑥 − 3 ⟹ 𝑅{𝑓} = (−3, ∞)
1 𝑥
2) 𝑓(𝑥 ) = (2) + 4 ⟹ 𝑅{𝑓} = (+4, ∞)
𝑥
3
3
3) 𝑓(𝑥 ) = −(√2) + 2 ⟹ 𝑅{𝑓} = (−∞, 2)
3 𝑥
2
5
7
4) 𝑓(𝑥 ) = − ( ) −
34
2
⟹ 𝑅{𝑓} = (−∞, − )
7
Laws of exponents: let 𝑎, 𝑏 ∈ 𝑅+ , then :
𝑥 𝑦
1) 𝑎 𝑎 = 𝑎
𝑥+𝑦
3) (𝑎 𝑥 )𝑦 = 𝑎 𝑥𝑦
𝑎𝑥
2) 𝑎𝑦 = 𝑎 𝑥−𝑦 , 𝑎 ≠ 0
4) (𝑎𝑏)𝑥 = 𝑎 𝑥 𝑏 𝑥
Example: Use the Laws of Exponential to rewrite and simplify
the expression.
8−4
a) 2−10
35
b) (𝑥 2 )−3 (4𝑥 −4 )2
Natural Exponential Function:
General Formula: 𝑓 (𝑥 ) = 𝑒 𝑥 𝑤ℎ𝑒𝑟𝑒 𝑒 ≈ 2.71
Example: Find the domain and range of the below functions.
1) 𝑓(𝑥 ) = 𝑒 𝑥 − 3
2) 𝑓(𝑥 ) = 𝑒 𝑥 + 3
3) 𝑓(𝑥 ) = −𝑒 𝑥 − 3
4) 𝑓(𝑥 ) = −𝑒 𝑥 + 3
36
Horizontal line test:
We say that the graph of a function is one-to-one if and only if
no horizontal line intersects its graph more than once
Example: Determine whether the below graphs of the functions
are one-to one function or not.
37
1)
2)
3)
4)
One-to-One functions:
𝑓(𝑥 ) = 𝑥 𝑛 =
𝑓(𝑥 ) 𝑖𝑠 1 – 1 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑
𝑓(𝑥 ) 𝑖𝑠 not 1 – 1 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛
Example:
One-to-one functions
Not one-to-one functions
𝑓(𝑥 ) = 𝑥 3 , 𝑓(𝑥 ) = 𝑥 5 + 3
𝑓(𝑥 ) = 𝑥 2 − 5 , 𝑓 (𝑥 ) = 5 − 𝑥 4
Example: Which one of the following is a one-to-one function.
a) 5 – 𝑥 4
38
b) 𝑒 𝑥
6
c) 𝑥 3 + 2
d) ln(𝑥 2 − 1)
Inverse Function:
Definition:
Example: If 𝑓(2) = 5 , then 𝑓 −1 (5) =
39
Cancellation equations:
𝑓 −1 (𝑓 (𝑥 )) = 𝑥 , 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑥 ∈ 𝑋
𝑓(𝑓 −1 (𝑥 )) = 𝑥 , 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑥 ∈ 𝑌
Example:
tan(tan−1 𝑥) = 𝑥
tan−1 (tan 𝑥) = 𝑥
log 𝑎 (𝑎 𝑥 ) = 𝑥
𝑎log𝑎 𝑥 = 𝑥
ln(𝑒 𝑥 ) = 𝑥
𝑒 ln 𝑥 = 𝑥
Finding the inverse of the function:
The graph of a function and the graph of its inverse are
symmetrical about the line 𝒚 = 𝒙
Ex) The graph of 𝑓 −1 is obtained by reflecting the graph of 𝑓 about
40
a) origin
b) x-axis
c) y-axis
d) the line 𝑦 = 𝑥
Example: Find the inverse function of the functions below:
1) 𝑓(𝑥 ) = 𝑥 3 + 2
2) 𝑓(𝑥 ) =
41
2𝑥−4
𝑥
Logarithmic Functions:
𝑓(𝑥 ) = log 𝑎 𝑥
, 𝑤ℎ𝑒𝑟𝑒 𝑎 ≠ 1 𝑎𝑛𝑑 𝑎 > 0
Note that:
𝑓(𝑥 ) = ln 𝑥 is called the Natural Logarithmic Function
Laws of Logarithmic:
If 𝑥, 𝑦 ∈ 𝑅+ 𝑎𝑛𝑑 𝑟 ∈ 𝑅 , 𝑡ℎ𝑒𝑛
1) log 𝑏 (𝑥𝑦) = log 𝑏 (𝑥) + log 𝑏 (𝑦)
𝑥
2) log 𝑏 (𝑦) = log 𝑏 (𝑥) − log 𝑏 (𝑦)
3) log 𝑏 (𝑥 𝑟 ) = 𝑟 log 𝑏 𝑥
4) log 𝑎 𝑎 = 1
,
log 𝑎 1 = 0
Example: Use the laws of logarithmic to compute
log 2 80 − log 2 5
42
Example: Solve the following equations:
1) 𝑒 5−3𝑥 = 10
2) log 3 (2𝑥 + 3) = 3
43
Inverse Trigonometric Functions:
Example: Evaluate
1
1) sin−1 (2)
√3
2) cos −1 ( 2 )
√3
3) sin−1 ( 2 )
44
Example: Find the domain of the function 𝑦 = sin−1 (3𝑥 + 1)
Solution:
45