Al-Farabi University College

Computer Engineering Department

Ass.Prof.Dr. Ihsan AlNaimi Engineering Analysis 2nd Class 1st Semester

2- Engineering Functions
2.1 Rational Functions: A rational function, R(x), has the form. )‫(الدوال النسبية‬
𝑷(𝒙)
R(x) =
𝑹(𝒙)

Where P and Q are polynomial functions; P is the numerator and Q is the denominator.
When sketching the graph of a rational function, 𝑦 = 𝑓(𝑥), it is usual to draw
up a table of x and y values. The figure below shows a graph of the function:
𝟏+𝟐𝒙 𝟏
𝒚= = + 𝟐 As x increases, the value of y approaches 2. We write this as
𝒙 𝒙
y → 2 as x → ∞ ‘y tends to 2 as x tends to infinity’
𝒚 → ±∞ as x → 0
As x → ∞, the graph gets nearer and nearer to the straight line y = 2. We say that y = 2
is an asymptote of the graph. Similarly, x = 0, that is the y axis, is an asymptote since the
graph approaches the line x = 0 as x → 0.

𝒙−𝟏
Ex.1 Sketch the rational function:
𝒙+𝟐
The asymptotes are the horizontal line y = 1 and the vertical line x = −2

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 Al-Farabi University College
Computer Engineering Department
Ass.Prof.Dr. Ihsan AlNaimi Engineering Analysis 2nd Class 1st Semester

2.2 Exponential Functions: ‫ الدوال األسية‬An exponent is another name for a power or
index. Expressions involving exponents are called exponential expressions, for
example 34, ab, and mn. In the exponential expression ax, a is called the base; x is the
exponent. Exponential expressions can be simplified and manipulated using the laws
of indices. These laws are summarized here:

𝒂𝟑𝒙 𝒂𝟐𝒙 𝒂𝒙+𝒚 𝒂𝒚

Ex.2 Simplify: 1- 2-
𝒂𝟒𝒙 𝒂𝟐𝒙
An exponential function, 𝑓(𝑥), has the form 𝒇(𝒙) = 𝒂𝒙 where a is a positive constant
called the base.
Some typical exponential functions are tabulated in the table below and are shown in
figure below. Note from the graphs that these are one-to-one functions. An exponential
function is not a polynomial function. The powers of a polynomial function are constants;
the power of an exponential function, that is the exponent, is the variable x.

The most widely used exponential function, commonly called the exponential function, is
𝒇(𝒙) = 𝒆𝒙 where e is an irrational constant ‫( ثابت أصم‬e = 2.718 281 828 . . .) commonly
called the exponential constant. This particular exponential function so dominates
engineering applications that whenever an engineer refers to the exponential function it
almost invariably means this one. As x increases positively, ex increases very rapidly;
as x → ∞, ex → ∞. As x increases negatively, ex approaches zero; as x → −∞, ex → 0.
Note that the exponential function is never negative.

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 Al-Farabi University College
Computer Engineering Department
Ass.Prof.Dr. Ihsan AlNaimi Engineering Analysis 2nd Class 1st Semester

𝒆𝟐𝒙 𝒆𝒙 (𝒆𝒙 +𝒆𝟐𝒙 )

Ex.2 Simplify: 1- 2-
𝟑𝒆𝟑𝒙 𝒆𝟐𝒙

2.3 Logarithm Functions: The equation 𝟏𝟔 = 𝟐𝟒 may be expressed in an alternative

form using logarithms. In logarithmic form we write log 2 16 = 4 and say ‘log to the
base 2 of 16 equals 4’. Hence logarithms are nothing other than powers. The
logarithmic form is illustrated by more examples:

In practice, most logarithms use base 10 or base e. Logarithms using base e are called
natural logarithms. log10 𝑥 and log 𝑒 𝑥 are usually abbreviated to log 𝑥 and ln 𝑥,
respectively.
Ex.3 Solve the equations: a- 𝟏𝟔 = 𝟏𝟎𝒙 b- 𝐥𝐨𝐠 𝒙 = 𝟏. 𝟓 c- 𝐥𝐧 𝒙 = 𝟎. 𝟕𝟓
Logarithmic expressions can be manipulated using the laws of logarithms. These laws
are identical for any base, but it is essential when applying the laws that bases are not
mixed.

𝟏 𝟒 𝟏
Ex.4 Simplify: a- 𝟑 𝐥𝐨𝐠 𝒙 + 𝐥𝐨𝐠 𝒙𝟐 b- 𝟓 𝐥𝐧 𝒙 + 𝐥𝐧 c- 𝐥𝐧(𝟐𝒙𝟑 ) − 𝐥𝐧 ( 𝟐) + 𝐥𝐧 𝟐𝟕
𝒙 𝒙 𝟑
a- 𝟑 𝐥𝐨𝐠 𝒙 + 𝐥𝐨𝐠 𝒙𝟐 = 𝐥𝐨𝐠 𝒙𝟑 + 𝐥𝐨𝐠 𝒙𝟐 = 𝐥𝐨𝐠 𝒙𝟓
𝟏 𝟏 𝒙𝟓
b- 𝟓 𝐥𝐧 𝒙 + 𝐥𝐧 = 𝐥𝐧 𝒙𝟓 + 𝐥𝐧 = 𝐥𝐧 = 𝐥𝐧 𝒙𝟒
𝒙 𝒙 𝒙
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 Al-Farabi University College
Computer Engineering Department
Ass.Prof.Dr. Ihsan AlNaimi Engineering Analysis 2nd Class 1st Semester

Ex.5 Find: 𝐥𝐨𝐠 𝟐 𝟏𝟒

𝐥𝐨𝐠 𝟏𝟒 𝟏.𝟏𝟒𝟔
𝐥𝐨𝐠 𝟐 𝟏𝟒 = = = 𝟑. 𝟖𝟎𝟕
𝐥𝐨𝐠 𝟐 𝟎.𝟑𝟎𝟏
The logarithm functions are defined by 𝒇(𝒙) = 𝐥𝐨𝐠 𝒂 𝒙 , 𝒙>𝟎
In particular the logarithm functions 𝒇(𝒙) = 𝐥𝐨𝐠 𝒙 and 𝒇(𝒙) = 𝐥𝐧 𝒙 are shown in the
figure below and some values are given in the table. The domain of both of these functions
is (0,∞) and their ranges are (−∞,∞).

The following properties should be noted:

Connection between exponential and logarithm functions:

In particular:

Ex.6 Plot the following function: 𝒚 = 𝟕𝒙

As x varies from -3 to 3, then y varies from 0.003 to 343, thus several of these points
would not be discernible on a graph; therefore this can be overcome by using a log scale.
𝒚 = 𝟕𝒙 → 𝐥𝐨𝐠 𝒚 = 𝐥𝐨𝐠 𝟕𝒙 = 𝒙 𝐥𝐨𝐠 𝟕 = 𝟎. 𝟖𝟒𝟓𝒙

A log-linear plot
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 Al-Farabi University College
Computer Engineering Department
Ass.Prof.Dr. Ihsan AlNaimi Engineering Analysis 2nd Class 1st Semester

2.4 The hyperbolic functions: ‫ الدوال الزائدية‬It is define 𝐜𝐨𝐬𝐡 𝒙 and 𝐬𝐢𝐧𝐡 𝒙 by
𝒆𝒙 + 𝒆−𝒙 𝒆𝒙 − 𝒆−𝒙
𝐲(𝐱) = 𝐜𝐨𝐬𝐡 𝒙 = , 𝐲(𝐱) = 𝐬𝐢𝐧𝐡 𝒙 =
𝟐 𝟐
−𝒙 𝒙 𝒙 −𝒙
𝒆 +𝒆 𝒆 −𝒆
𝐜𝐨𝐬𝐡(− 𝒙) = = 𝐜𝐨𝐬𝐡 𝒙 , 𝐬𝐢𝐧𝐡(−𝒙) = = − 𝐬𝐢𝐧𝐡 𝒙
𝟐 𝟐
𝐬𝐢𝐧𝐡 𝒙 𝐜𝐨𝐬𝐡 𝒙
𝐭𝐚𝐧𝐡 𝒙 = , 𝐜𝐨𝐭𝐡 𝒙 =
𝐜𝐨𝐬𝐡 𝒙 𝐬𝐢𝐧𝐡 𝒙
𝟏 𝟏
𝐜𝐬𝐜𝐡 𝒙 = , 𝐬𝐞𝐜𝐡 𝒙 =
𝐬𝐢𝐧𝐡 𝒙 𝐜𝐨𝐬𝐡 𝒙
So, for example, 𝑐𝑜𝑠ℎ 1.7 = 𝑐𝑜𝑠ℎ (−1.7) and 𝑠𝑖𝑛ℎ (−1.7) = −𝑠𝑖𝑛ℎ 1.7, hyperbolic
functions are nothing other than combinations of the exponential functions 𝑒 𝑥 and 𝑒 −𝑥 .
However, these particular combinations occur so frequently in engineering that it is worth
introducing the 𝑐𝑜𝑠ℎ 𝑥 and 𝑠𝑖𝑛ℎ 𝑥 functions. The remaining hyperbolic functions are
defined in terms of 𝑐𝑜𝑠ℎ 𝑥 and 𝑠𝑖𝑛ℎ 𝑥.
Hyperbolic identities: Several identities involving hyperbolic functions exist. They can
be verified algebraically using the definitions given, and are listed for reference.

Ex.7 Express these equations: a- 𝟑𝒆𝒙 − 𝟐𝒆−𝒙 in terms of 𝑐𝑜𝑠ℎ 𝑥 and 𝑠𝑖𝑛ℎ 𝑥
b- 𝟐𝒔𝒊𝒏𝒉 𝒙 + 𝒄𝒐𝒔𝒉 𝒙 in terms of ex and e−x
a- 𝟑𝒆𝒙 − 𝟐𝒆−𝒙 = 𝟑(𝒄𝒐𝒔𝒉 𝒙 + 𝒔𝒊𝒏𝒉 𝒙) − 𝟐(𝒄𝒐𝒔𝒉 𝒙 − 𝒔𝒊𝒏𝒉 𝒙) = 𝒄𝒐𝒔𝒉 𝒙 + 𝟓 𝒔𝒊𝒏𝒉 𝒙
𝒆𝒙 +𝒆−𝒙 𝟑𝒆𝒙 −𝒆−𝒙
b- 𝟐𝒔𝒊𝒏𝒉 𝒙 + 𝒄𝒐𝒔𝒉 𝒙 = 𝒆𝒙 − 𝒆−𝒙 + =
𝟐 𝟐

2.4.1 Inverse hyperbolic functions: The inverse of the function 𝒔𝒊𝒏𝒉𝒙 is denoted by
𝐬𝐢𝐧𝐡−𝟏 𝒙. Here the −1 must not be interpreted as a power but rather the notation we use
for the inverse function. Similarly the inverses of 𝒄𝒐𝒔𝒉 𝒙 and 𝒕𝒂𝒏𝒉 𝒙 are denoted by
𝐜𝐨𝐬𝐡−𝟏 𝒙 and 𝐭𝐚𝐧𝐡−𝟏 𝒙 respectively.
Ex.8 Evaluate the following: a- 𝒄𝒐𝒔𝒉 𝟏. 𝟔 b- 𝒔𝒊𝒏𝒉 − 𝟐 c- 𝐬𝐢𝐧𝐡−𝟏 𝟑 d- 𝐭𝐚𝐧𝐡−𝟏 (−𝟎. 𝟐𝟓)
Ex.9 Express: 𝒂 𝒄𝒐𝒔𝒉 𝒙 + 𝒃 𝒔𝒊𝒏𝒉 𝒙, where a and b are constants in terms of 𝑒 𝑥 and 𝑒 −𝑥

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 Al-Farabi University College
Computer Engineering Department
Ass.Prof.Dr. Ihsan AlNaimi Engineering Analysis 2nd Class 1st Semester

Ex.10 Prove the following: a- 𝐜𝐨𝐬𝐡𝟐 𝒙 − 𝐬𝐢𝐧𝐡𝟐 𝒙 = 𝟏 b- 𝟏 − 𝐭𝐚𝐧𝐡𝟐 𝒙 = 𝒔𝒆𝒄𝒉𝟐 𝒙

𝟐 𝟐 𝒆𝒙 +𝒆−𝒙 𝟐 𝒆𝒙 −𝒆−𝒙 𝟐 𝒆𝟐𝒙 +𝟐+𝒆−𝟐𝒙 𝒆𝟐𝒙 −𝟐+𝒆−𝟐𝒙 𝟒

a- 𝐜𝐨𝐬𝐡 𝒙 − 𝐬𝐢𝐧𝐡 𝒙 = ( ) −( ) = − = =𝟏
𝟐 𝟐 𝟒 𝟒 𝟒

b- We start with the identity proved in (a): cosh2x – sinh2x = 1

then divide both sides by cosh2x, we get:
𝟏
𝟏 − 𝐭𝐚𝐧𝐡𝟐 𝒙 = = 𝐬𝐞𝐜𝐡𝟐 𝒙
𝐜𝐨𝐬𝐡𝟐 𝒙
𝝅 𝝅
Ex.11 Plot: a- 𝒚 = 𝐬𝐢𝐧 𝒙 b- 𝒚 = 𝐜𝐨𝐬 𝒙 c- 𝒚 = 𝐭𝐚𝐧 𝒙 d- 𝒚 = 𝐬𝐢𝐧−𝟏 𝒙, − ≥ 𝒚 ≤
𝟐 𝟐

Ex.12 Plot: a- 𝒚 = 𝐬𝐢𝐧𝐡 𝒙 b- 𝒚 = 𝐜𝐨𝐬𝐡 𝒙 c- 𝒚 = 𝐭𝐚𝐧𝐡 𝒙

a- b-
Note that sinh has domain ℝ and range ℝ, whereas cosh has domain ℝ and range {1,∞}

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 Al-Farabi University College
Computer Engineering Department
Ass.Prof.Dr. Ihsan AlNaimi Engineering Analysis 2nd Class 1st Semester

c- 𝐭𝐚𝐧𝐡 𝒙 has the horizontal asymptotes 𝑦 = ±1

Ex.13 Plot: a- 𝒚 = 𝐬𝐢𝐧𝐡−𝟏 𝒙 b- 𝒚 = 𝐜𝐨𝐬𝐡−𝟏 𝒙 c- 𝒚 = 𝐭𝐚𝐧𝐡−𝟏 𝒙

a-

b-

c-