D. T. PHAM - VGU Chapter 1: FUNCTIONS AND GRAPHS Duong T. PHAM - EEIT2014 Fundamental Engineering Mathematics for EEIT2014 Vietnamese German University Binh Duong Campus Duong T. PHAM - EEIT2014 September 24, 2014 1 / 55
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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Increasing and Decreasing Functions
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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Increasing and Decreasing Functions
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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Vertical and horizontal stretching and reflecting
Ex:
x
y
1
y = cos x2
y = 2 cos x
1/2
y = (0.5) ∗ cos x
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Vertical and horizontal stretching and reflecting
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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