fundamental engineering maths 1

D.T.PHAM

- VGUChapter 1: FUNCTIONS AND GRAPHS

Duong T. PHAM - EEIT2014

Fundamental Engineering Mathematics for EEIT2014

Vietnamese German UniversityBinh Duong Campus

Duong T. PHAM - EEIT2014 September 24, 2014 1 / 55

D.T.PHAM

- VGU

Outline

1 Functions

2 Mathematical Models and Essential Functions

3 New functions from old functions

4 Inverse functions


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Example 1

r

A circle of radius r

Area of the circle: A = πr2

Each number r , there is one and only one number A (A = πr2)

Say: A is a function of r


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Example 2

The world population grows as in the following table

Year Population (Millions)

1750 791

1800 978

1850 1,262

1900 1,650

1950 2,519

2000 6,070

1914 7,257

The world population P depends on time t (year). For example,P(1800) = 978× 106; P(2000) = ? 6070× 106

Each value time t, there is one and only one value of populationP(t).

Say: P is a function of t


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Definition of functions

Definition

A function f is a rule that assigns to each element x in a set D exactlyone element, called f (x), in a set E .

The set D is called the domain of the function f ;

The number f (x) is the value of f at x ;

The range of f is

Range of f = {f (x) : x ∈ D}

The symbol x is called the independent variable of f ;


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Example 2

The world population

t P

1750 791

1800 978

1850 1,262

1900 1,650

1950 2,519

2000 6,070

1914 7,257

The domain D of P is ?

D = {1750, 1800, 1850, 1900, 1950, 2000, 1914}

The range of P is ?Range of P = {P(1750),P(1800),P(1850),P(1900),P(1950),P(2000),P(1914)}={791, 978, 1262, 1650, 1650, 2519, 6070, 7257}

The symbol t is the independent variable ;

The symbol P is the dependent variable


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Diagrams for a function

Machine diagram:

x f (x)f(Input) (Output)

Arrow diagram:

x f (x)

a f (a)

D Ef


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Graph

Definition

If f is a function with domain D, then its graph is the set of ordered pairs{(x , f (x)

): x ∈ D

}Ex: Plot the graph of the function f : [0, 2]→ R defined byf (x) = x2 − x + 1

0 x

y

0.5 1 2

0.751

3


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Representations of Functions

There are 4 ways to represent a functions:

By a description by words (verbally)

By a table of values (numerically)

By a graph (visually)

By an explicit formula (algebraically)


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Example of Circle Area

Area of a circle :

Verbally: The area of a circle is equal to square of the radiusmultiplying with π

Algebraically: A = πr2

Visually:

r

A

A = πr2

r


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Example of World Population

Numerical description

t P

1750 791

1800 978

1850 1,262

1900 1,650

1950 2,519

2000 6,070

1914 7,257


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Another Example

Ex: We need a rectangular storage container with an open top which hasa volume of 10m3. The length of its base is required to be twice its width.Material for the base costs $10/m2; material for the sides costs $6/m2.Express the cost of materials as a function of the width of the base.

w

2w

h


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Another Example

w

2w

h

Area of the base = w ∗ (2w) = 2w2

⇒ Cost = 10 ∗ (2w2) = 20w2$

Area of the front and back sides= 2 ∗ (2hw) = 4hw

Area of the left and right sides= 2 ∗ (hw) = 2hw

⇒ Area of 4 sides = 6hw ⇒ Cost = 6 ∗ (6hw) = 36hw$

⇒ Total cost = 20w2 + 36hw($)

Volume = 10 ⇒ 2hw2 = 10 ⇒ h = 5/w2

⇒ Total cost: C (w) = 20w2 + 180w


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Domain of a function

Remark: If a function is given by a formula and the domain is not statedexplicitly, the convention is that the domain is the set of all numbers forwhich the formula makes sense and defines a real number .

Ex: Find the domain of the function f (x) =√x + 1.

Ans: The formula√x + 1 is well-defined when x + 1 ≥ 0, which is

equivalent to x ≥ −1.

The domain of the above function is

D = [−1,+∞)


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The Veritcal Line Test

The graph of a function is a curve in xy -plane.

Question : Which curves in the xy -plane are graphs of functions?

Vertical Line Test: A curve in the xy -plane is the graph of a function ofx if and only if no vertical line intersects the curve more than once.

x

y

a

x

y

a

b

c

f (a) = b or f (a) = c?


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Piecewise defined functions

Ex: A function f is defined by: f (x) =

{1− x if x ≤ 1

x2 if x > 1.

f (−1) = ? 1− (−1) = 2f (2) =? 22 = 4f (1) =? 1− 1 = 0

x

y

1

1

−1 2

2

4


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Piecewise defined functions

Ex: Sketch the graph of the absolute value function f (x) = |x |

Ans: We have

f (x) = |x | =

{x if x ≥ 0

−x if x < 0

x

y


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Symmetry: Even Functions

x

y

1−1 2−2

1

4

Graph of function f (x) = x2

f (−1) =? (−1)2 = 1;f (1) =? 12 = 1;⇒ f (−1) = f (1)

f (−2) = (−2)2 = 4 andf (2) = 22 = 4⇒ f (−2) = f (2)

f (−x) = (−x2) = x2

f (x) = x2 ⇒ f (−x) = f (x)

f is an even function

Definition

A function f : D → R is said to be even if f (−x) = f (x) ∀x ∈ D


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Symmetry: Odd Functions

x−x

f (x)

f (−x)

f (−x) = −f (x) ∀x ∈ D and f is said to be an odd function

Definition

A function f : D → R is said to be odd if f (−x) = −f (x) ∀x ∈ D


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Symmetry: Examples

Ex: Determine whether each of the following functions is even, odd, orneither even nor odd

f (x) = x3 − x ; g(x) = 1 + x2; h(x) = x + 1.

Ans:

(i) f (−x) = (−x)3 − (−x) = −x3 + x = −(x3 − x) = −f (x)⇒ f is an odd function.

x−x

f (x)

f (−x)


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Symmetry: Examples

Ans: g(x) = 1 + x2;

g(−x) = 1 + (−x)2 = 1 + x2 = g(x)⇒ g is an even function.

x−x

g(x)g(−x)


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Symmetry: Examples

Ans: h(x) = 1 + x ;

h(−x) = 1 + (−x) = 1− x ⇒ h(−x) 6= h(x) and h(−x) 6= −h(x)⇒ h is NOT either even or odd.

x

y


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Increasing and Decreasing Functions

A

B

C

D

a b c d x

y

x1 x2

f (x1)

f (x2) y = f (x)

The graph of f rises between A and B, falls between B and C , andrises again between C and D.

We say that: f is increasing on the intervals [a, b] and [c , d ]f is decreasing on the intervals [b, c]


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A

B

C

D

a b c d x

y

x1 x2

f (x1)

f (x2) y = f (x)

f is increasing on [a, b]; Suppose a ≤ x1 < x2 ≤ b ⇒ f (x1) < f (x2)

A function f is called increasing on an interval I if

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

A function f is called decreasing on an interval I iff (x1) > f (x2) whenever x1 < x2 in I


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Ex:

x

y

0

y = x2

f (x) = x2 is decreasing on (−∞, 0] because if x1 < x2 ≤ 0, then|x1| > |x2| ≥ 0 and f (x1) = x21 = |x1|2 > |x2|2 = x22 = f (x2)

f (x) = x2 is increasing on [0,∞) because if 0 ≤ x1 < x2, thenx21 < x22 , which means f (x1) < f (x2)


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Exercises

I.1: 1–9; 17; 19–22; 27–33;36; 40; 43; 45–50;52; 57; 61–62; 64; 65–70


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Mathematical Model

Potential questions:

What is the area of the yellow domain?

What is the volume of the wing?

and many other essential questions


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Mathematical Models

Real–worldproblems

Mathematicalproblems

Mathematicalconclusions

Real–worldpredictions

Formulae

Test

SolveInterpret


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Linear Models

Def: A function y = f (x) is linear if its graph is a straight line. Theformula of a linear function has the following formula

y = ax + b,

where a is the slope of the line and b is the y -intercept.

Ex: y = −12x + 2

0 x

y

4

2


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Polynomials

Def: A function P is called a polynomial if

P(x) = anxn + an−1x

n−1 + . . .+ a1x + a0

where

n: nonnegative integer,

a0, a1, . . . , an are coefficients

The domain of P is R = (−∞,∞). If an 6= 0, the degree of P is n.

Ex:

A linear function y = ax + b is a polynomial of degree 1,

A polynomial of degree 2 has the form y = ax2 + bx + c and iscalled a quadratic function,

A polynomial of degree 3 has the form y = ax3 + bx2 + cx + d andis called a cubic function.


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Power Functions

A function of the form y = xα where α is a constant is called a powerfunction.

Remark: Note here that α is a real number.

If α = n where n is a positive integer, then y = xn is a polynomialof degree n;The domain of y = xn is R.

If α = 1/n where n is a positive integer, then y = x1/n is called aroot function.Note: y = n

√x ⇒ yn = x and

domain of y = n√x is

{R+ = [0,∞) if n is even

R if n is odd


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Power Functions

y = n√x

y =√x

x

yy = 3√x

x

y

y = x−1 = 1x ; domain is R\{0}


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Rational Functions

Def: A rational function f is a ratio of two polynomials:

f (x) =P(x)

Q(x)

where P and Q are polynomials.

Domain of f isD = {x ∈ R : Q(x) 6= 0}

Ex: f (x) =x

x2 − 3x + 2is a rational function and the domain

D = {x ∈ R : x2 − 3x + 2 6= 0}= {x ∈ R : x 6= 1 and x 6= 2}


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Algebraic Functions

Def: A function f is called an algebraic function if it can be constructedusing algebraic operations (such as addition, subtraction, multiplication,division, and taking roots) starting with polynomials

Ex:

f (x) = anxn + . . .+ a1x + a0 and g(x) = bmx

m + . . .+ b1x + b0 arealgebraic functions

f + g , f − g , f ∗ g , f /g and k√f are algebraic functions

h(x) = 1k√xn+1

is an algebraic function


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Trigonometric Functions

y = sin x

−2π

2π−π π

0

1

−1

y = cos x

−2π

2π−π π

0

1

−1


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Exponential Functions

Def: Exponential functions are the functions of the form f (x) = ax

where a > 0.

The domain = RThe range = {positive numbers}

x

y

y = 2x

x

y

y = ( 12 )x


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Logarithmic Functions

Def: The logarithmic functions f (x) = loga x , where the base a is a posi-tive constant, are the inverse functions of the exponential functions.

Domain = (0,∞)

Range = (−∞,∞)

x

y

y = log2 x


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Transcendental Functions

Def: Transcendental functions are functions that are NOT algbraicfunctions.

Ex:

Trigonometric functions and their inverses are transcendentalfunctions,

Exponential and logarithmic functions are transcendental functions.

Ex: Classify the following functions as one of the types of functions thatwe have discussed.

1 3x −→ exponential function,

2 x5 −→ power function (or polynomial of degree 5)

3 1+x1−√x−→ algebraic function,

4 1− x − x4 −→ polynomial of degree 4.


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Exercises

I.2:

1-2; 5, 6, 8, 9; 10

16, 19, 20


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Transformation of functions

Vertical and horizontal shifts: Suppose c > 0. To obtain the graphof

y = f (x) + c , shift the graph of y = f (x) a distance c unitsupward

y = f (x)− c , shift the graph of y = f (x) a distance c unitsdownward

y = f (x + c) , shift the graph of y = f (x) a distance c units tothe left

y = f (x − c) , shift the graph of y = f (x) a distance c units tothe right


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Vertical and horizontal shifts

y = f (x)

c

c

c

c

y = f (x) + c

y = f (x)− c

y = f (x + c) y = f (x − c)


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Vertical and horizontal stretching and reflecting

Vertical and horizontal Stretching and Reflecting: Suppose c > 1.To obtain the graph of

y = cf (x) , stretch the graph of y = f (x) vertically by a factor of c

y = (1/c)f (x) , compress the graph of y = f (x) vertically by afactor of c

y = f (cx) , compress the graph of y = f (x) horizontally by afactor of c

y = f (x/c) , stretch the graph of y = f (x) horizontally by a factorof c

y = −f (x) , reflect the graph of y = f (x) about the x-axis

y = f (−x) , reflect the graph of y = f (x) about the y -axis


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Ex:

x

y

1

y = cos x2

y = 2 cos x

1/2

y = (0.5) ∗ cos x


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Ex:

x

y

1

y = cos x

y = cos 2x

y = cos 12x


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Combinations of functions

Given f : A→ R and g : B → R, we have:

f ± g : A ∩ B → R defined by

(f ± g)(x) = f (x)± g(x), x ∈ A ∩ B

f ∗ g : A ∩ B → R defined by

(f ∗ g)(x) = f (x) ∗ g(x), x ∈ A ∩ B

f ± g : {x ∈ A ∩ B : g(x) 6= 0} → R defined by(f

g

)(x) =

f (x)

g(x), x ∈ {x ∈ A ∩ B : g(x) 6= 0}


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Combinations of functions

Ex: Given f (x) =√x and g(x) =

√2− x .

Domain of f is ? [0,∞); Domain of g is ? (−∞, 2]

=⇒ [0,∞) ∩ (−∞, 2] = [0, 2]

(f ± g) : [0, 2]→ R given by (f ± g)(x) =√x ±√

2− x

fg : [0, 2]→ R given by (fg)(x) =√x√

2− x

fg : [0, 2)→ R given by

(f

g

)(x) =

√x√

2− x


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Composition of functions

Def: Let f : A→ B and g : B → C . The composition of f and g isdefined by

gf : A −→ Cx 7−→ g(f (x))

Ex: f (x) = x2 and g(x) = 2− 3√x

We have

gf (x) = g(f (x)) = g(x2) = 2− 3√x2

and

fg(x) = f (g(x)) = f (2− 3√x) = (2− 3

√x)2


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Exercises

I.3:

1, 2, 5, 9–14

29–36, 37–38

41, 42

50,51

61,62


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One-to-one functions

a 1b 2c 3d 4

D Ef

a 1b 2c 3d 4

D Eg

g(b) = g(c)

f is one-to-one and g is NOT one-to-one

Def: A function f is called a one-to-one function if

f (x1) = f (x2) =⇒ x1 = x2


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Horizontal line Test

Ex:

x

y

y = f (x)

x1 x2

f (x1)

f (x2)

||

f is not one-to-one

Horizontal line test: A function is one-to-one if and only if NOhorizontal line intersects its graph more than once.


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Inverse function

Def: Let f be a one-to-one function with domain A and range B. Thenits inverse function f −1 has domain B and range A and is defined by

f −1(y) = x ⇐⇒ f (x) = y ∀y ∈ B.

Ex:

a 1b 2c 3d 4

D Ef

a1b2c3d4

E Df −1

f −1 ◦ f (a) = ? f −1(f (a)) = f −1(1) = a

f ◦ f −1(3) = ? f (f −1(3)) = f (c) = 3.


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Cancellation equations

Let f be a one-to-one function with domain A and range B.

f −1(f (x)) = x ∀x ∈ A

f (f −1(y)) = y ∀y ∈ B

x f (x)f xf −1


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Find an inverse

To find the inverse of a one-to-one function f :

Step 1: Write y = f (x)

Step 2: Solve this equation for x in terms of y

Step 3: To obtain f −1 as a function of x , interchange x and y .The resulting equation is y = f −1(x).

Ex: Find the inverse of y = 3x3 + 5Ans:

y = 3x3 + 5

=⇒ 3x3 = y − 5 =⇒ x = 3

√y−53

y = 3

√x−53 is the inverse of y = 3x3 + 5.


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Graphs of inverse functions

Suppose f (a) = b ⇒(a, b) ∈graph of f

⇒f −1(b) = a ⇒ (b, a) ∈ graphof f −1

The graph of f −1 is symmetricto that of f

x

y

y = f (x)

a

b

a

b

y = f−1(x)

The graph of f −1 is obtained by reflecting the graph of f about the liney = x


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Exercises

I.6:

3–12; 15–18; 21–24


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