3-1: functions; domain and rangeuowa.edu.iq/filestorage/file_1550831554.pdf · 3-1: functions;...

Chapter three The functions Asst. Lec. Ahmed Oleiw

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3-1: Functions; Domain and Range

The area of a circle depends on the radius of the circle. The distance an object

travels at constant speed along a straight-line path depends on the elapsed time...etc.

In each case, the value of one variable quantity, say y, depends on the value of

another variable quantity, which we might call x. We say that "y is a function of x"

and write this symbolically as

y = f(x)

The symbol f represents the function, the letter x is the independent variable

representing the input value of f, and y is the dependent variable or output value of

f at x as shown in the figure (3.1):

Figure (3.1): The function diagram.

DEFINITION:

A function f from a set D (domain) to a set Y (range) is any rule that assigns a

unique (single) element f(x) ∈ Y to each element x ∈ D as shown in the figure (3.2):

Figure (3.2): A function from a set D to a set Y assigns a

unique element of Y to each element in D

Example (1): Find the domain and range of each function:

a- 𝑦 = √𝑥 + 4 b- 𝑦 =1

𝑥−2 c- 𝑦 = √9 − 𝑥2 d- 𝑦 = √2 − √𝑥


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Solution:

a- 𝑥 + 4 ≥ 0 → 𝑥 ≥ −4

∴ Df = {x: x ∈ R, x ≥ −4} Rf = {y: y ∈ R, y ≥ 0}

b- 𝑥 − 2 ≠ 0 → 𝑥 ≠ 2

∴ Df = {x: x ∈ R, x ∀ R/2}

𝑥 =1

𝑦+ 2

Rf = {y: y ∈ R, y ∀ R/0}

c- 9 − 𝑥2 ≥ 0 → 𝑥2 ≥ 9 − 3 ≤ 𝑥 ≤ 3 ∴ Df: −3 ≤ 𝑥 ≤ 3

∴ Rf: 0 ≤ 𝑦 ≤ 3

d- 2 − √𝑥 ≥ 0 → 0 ≤ 𝑥 ≤ 4 → Df: 0 ≤ 𝑥 ≤ 4

If 𝑥 = 0 → 𝑦 = √2

If 𝑥 = 4 → 𝑦 = 0

Rf: 0 ≤ y ≤ √2

Note: any polynomial of the following forms has the domain R. For example:

f(x) =1

2x3 + 3x2 − x + π

f(x) = 5x2 − 2x − √2 ≫≫≫≫≫≫≫≫ Df = R

f(x) =3

2x5 + x3 − x + 1

3-2: Inequalities:

Rules for Inequalities:

If a, b and c are real numbers, then:

1. a < b → a + c < b + c

2. a < b → a − c < b − c

3. a < b and c > 0 → ac < bc

4. a < b and c < 0 → bc < ac

Special case: a < b → −b < −a

5. a > 0 and 1

a> 0

6. If a and b are both positive or both negative, the a < b →1

b<

1

a .


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Intervals:

- The open interval is the set of all real numbers that be strictly between two fixed

numbers a and b:

(a, b)≡ a < x < b

- The closed interval is the set of all real numbers that contain both endpoints:

[a, b ] ≡ a ≤ x ≤ b

- Half open interval is the set of all real numbers that contain one endpoint but not

both:

[a, b ) ≡ a ≤ x < b

(a,b ] ≡ a < x ≤ b

Example (2): Solve the following inequalities and show their solution sets on the real

line. (𝑎) 2𝑥 − 1 < 𝑥 + 3, (𝑏) −𝑥

3< 2𝑥 + 1, (𝑐)

6

𝑥−1≥ 5

Solution:

(𝑎) 2𝑥 − 1 < 𝑥 + 3 → 𝑥 < 4

The solution set is the open interval (-∞,4)

(b) −𝑥

3< 2𝑥 + 1

−𝑥 < 6𝑥 + 3 → −7𝑥 < 3 → 𝑥 >−3

7

(c) 6

𝑥−1≥ 5 → 6 ≥ 5𝑥 − 5 → 11 ≥ 5𝑥 →

11

5≥ 𝑥.


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Absolute Value:

The absolute value of a number x, denoted by|𝑥|, is defined by the formula

Absolute value properties

Example (3): Solving an equation with absolute values |2𝑥 − 3| = 7

Solution:


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Example (4): Solving an inequality involving absolute values|𝟓 −𝟐

𝒙| < 𝟏.

Solution:

Example (5): Solve the inequality and show the solution set on the real line:

Solution:


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3-3: Composition of functions:

Suppose that the outputs of a function f can be used as inputs of a function g.

We can then hook f and g together to form a new function whose inputs are the inputs

of f and whose outputs are the numbers:

(gof)(x) = g(f(x))

Example (6): Let f(x) =x

x−1 and g(x) = 1 +

1

x . Find (gof)(x) and (fog)(x).

Solution:

(gof)(x) = g(f(x)) = g (x

x − 1) = 1 +

1x

x − 1

=2x − 1

x

(fog)(x) = f(g(x)) = f (1 +1

x) =

1 +1x

1 +1x

− 1= x + 1

Example (7): Let (gof)(x) = x and f(x) =1

x . Find g(x).

Solution: (gof)(x) = g (1

x) = 𝐱 → g(x) =

1

x

3-4: Increasing and Decreasing Functions:

If the graph of a function rises as you move from left to right, we say that the

function is increasing. If the graph descends or falls as you move from left to right,

the function is decreasing.

DEFINITION:

Let f be a function defined on an interval I and let x1, and x2 be any two points

in I.

1. If f(x2) > f (x1) whenever x1 < x2, then f is said to be increasing on I.

2. If f(x2) < f(x1) whenever x2 < x1, then f is said to be decreasing on I.

3-5: Even Functions and Odd Functions:

The graphs of even and odd functions have chamcteristic symmetry properties.

DEFINITION:

A function y = f(x) is an

Even function of x if f (-x) = f(x),


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Odd function of x if f (-x) = - f(x),

The graph of an even function is symmetric about the y-axis. Since f (-x) =

f(x), a point (x, y) lies on the graph if and only if the point (-x, y) lies on the graph

see figure (3.3-a).

The graph of an odd function is symmetric about the origin. Since f (-x) = - f(x),

a point (x, y) lies on the graph if and on1y if the point (-x, -y) lies on the graph see

figure (3.3-b).

Example (8):

f(x) = x2 Even function: (−x)2 = x2 for all x; symmetry about y- axis.

f(x) = x2 + 1 Even function: (−x)2 + 1 = x2 + 1 for all x; symmetry about y-

axis.

f(x) = x Odd function: (−x) = −x for all x; symmetry about origin.

f(x) = x + 1 Not odd:f(−x) = −x + 1, but−f(x) = −x − 1. The two are not

equal.

Figure (3.3): (a) The graph of y = x2 (an even function) is symmetric about the y-axis.

(b) The graph of y = x3 (an odd function) is symmetric about the origin.


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3-6: Types of Functions:

There are a number of important types of functions frequently encountered in

calculus. We will summarize them here.

1. Linear Functions: A function of the form 𝒇(𝒙) = 𝒎𝒙 + 𝒃 for constants m

and b is called a linear function. Figure (4-a) shows an array of lines f(x) =

mx where b = 0 so these lines pass through the origin. Constant functions result

when the slope m= 0 see figure (4-b).

2. Power Functions: A function 𝒇(𝒙) = 𝒙𝒂 where a is a constant, is called a

power function. The graphs of 𝑓(𝑥) = 𝑥𝑛for n=1, 2, 3, 4, 5, are displayed in

figure (5).

Figure (5): The graphs of 𝑓(𝑥) = 𝑥𝑛for n=1, 2, 3, 4, and 5.

3. Polynomials: A function p is a polynomial if

𝒑(𝒙) = 𝒂𝒏𝒙𝒏 + 𝒂𝒏−𝟏𝒙𝒏−𝟏 + ⋯ + 𝒂𝟏𝒙 + 𝒂𝟎

Figure (4-a): The collection of lines y=mx has

slope m and all lines pass through the

origin.

Figure (4-b): A constant function has slope m = 0.


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Where n is a nonnegative integer and the numbers 𝑎0, 𝑎1𝑎2, … . , 𝑎𝑛 are real

constants (called the coefficients of the polynomial) and n is called the degree of the

polynomial. Linear functions with 𝑚 ≠ 0 are polynomials of degree 1. Polynomials

of degree 2, usually written as 𝑝(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐, are called quadratic functions.

Likewise, cubic functions are polynomials 𝑝(𝑥) = 𝑎𝑥3 + 𝑏𝑥2 + 𝑐𝑥 + 𝑑 see figure

(6).

Figure (6): Shows the graphs of three polynomials.

4. Rational Functions: is a ratio of two polynomials:

𝒇(𝒙) =𝒑(𝒙)

𝒒(𝒙)

For example, the function f(x) =2x2−3

7x+4

Figure (7): Graphs of three rational functions.


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5. Algebraic Functions: An algebraic function is a function constructed from

polynomials using algebraic operations (addition, subtraction, multiplication,

division, and taking roots). Rational functions are special cases of algebraic

functions see figure (8).

Figure (8): Graphs of three algebraic functions.

6. Trigonometric Functions: The graphs of the sine and cosine functions are

shown in figure (9).

Figure (9): Graphs of the sine and cosine functions.

7. Exponential Functions: Functions of the form (𝐱) = 𝐚𝐱 , where the base 𝑎 >

0, is a positive constant and a ≠ 1, are called exponential functions. All

exponential functions have domain (−∞, ∞)and range (0, ∞) see figure (10).


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Figure (10): Graphs of exponential functions.

8. Logarithmic Functions: These are the functions 𝒇(𝒙) = 𝐥𝐨𝐠𝒂 𝒙 , where the

base 𝒂 ≠ 𝟏 is a positive constant. They are the inverse functions of the

exponential functions see figure (11). In each case the domain is (0, ∞) and

the range is(−∞, ∞).

Figure (11): Graphs of Logarithmic functions.


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3-7: Trigonometric Functions:

This section reviews radian measure and the basic trigonometric functions.

Radian Measure:

In navigation and astronomy, angles are measured in degrees, but in calculus it

is best to use units called radians because of the way they simplify later calculations.

The radian measure of the angle ACB at the center of the unit circle as shown

in figure (12) equals the length of the arc that ACB cuts from the unit circle.

Figure (12): Unit circle.

Thus [s= r Ө] is the length of arc on a circle of radius r when Ө is measured in

radians.

If the circle is a unit circle having radius r = 1, then from Figure (12) and the above

equation, we see that the central angle Ө measured in radians is just the length of the

arc that the angle cuts from the unit circle. Since one complete revolution of the unit

circle is 360° or 2π radians, we have


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Angles measured counterclockwise from the positive x-axis are assigned

positive measures; angles measured clockwise are assigned negative measures as

shown in figure (13).

The Six Basic Trigonometric Functions


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The CAST rule, remembered by the statement “All Students Take Calculus,”

tells which trigonometric functions are positive in each quadrant.

The coordinates of any point P(x, y) in the plane can be expressed in terms of

the point's distance r from the origin and the angle Ө that ray OP makes with the

positive x-axis. Since 𝑥𝑟⁄ = cos 𝜃 𝑎𝑛𝑑

𝑦𝑟⁄ = sin 𝜃, we have

When r = 1 we can apply the Pythagorean Theorem to the reference right triangle and

obtain the equation:

Dividing this identity in turn by 𝑐𝑜𝑠2𝜃, 𝑠𝑖𝑛2𝜃 gives:


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Addition & Subtract Formulas:

Double Angle Formulas:

Half Angle Formulas:

Periodicity:

Definition: A function f(x) is periodic if there is a positive number p such that f(x +

p) = f(x) for every value of x. The smallest such value of p is the period

of f.


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Symmetry:

Shift Formulas:

sin(𝑥 ±𝜋

2) = ± cos(𝑥)

cos(𝑥 ±𝜋

2) = ∓ sin(𝑥)


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Example (9): Prove the following identities:

Solution:

Example (10): Simplify

Solution:

Solution:

Example (11):

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