Chapter 1 (functions).

BMM 104: ENGINEERING MATHEMATICS I of 13

CHAPTER 1: FUNCTIONS

Relations

Types of relations :

One – to – one

one – to – many

many – to – one

Ordered Pairs

f(x ) ( a , c ) and ( b , d ) are known as ordered pairs . The set of ordered pairs is { ( a , c ) , ( b , d ) } . c and d are called the image of the corresponding

first component .domain codomain

Functions

.a

.b

.c

.d


Definition Function A relation in which every element in the domain has a unique image in the codomain.

Notation of functions :A function f from x to y : f : x y or y = f ( x )

f(x )

domain codomain

Domain – set of input values for a function Range – the corresponding output values – is a subset of codomain

Elements of domain { a , b }Elements of codomain { c , d }

NOTE: Vertical line test can be used to determine whether a relation is a function or not. A function can have only one value for each x in its domain, so no vertical line can intersect the graph of a function more than once.

Example: Determine which of the following equations defines a function y in terms of x. Sketch its graph.

(i) (ii) (iii)

Domain and Range

The set D of all possible input values is called the domain of the function.The set of all values of as x varies throughout D is called the range of the function.

Example:

Combining Functions

.a

.b

.c

.d


Sum, Differences, Products, and Quotients

Example: Attend lecture.

Composite Functions

If f and g are functions, the composite function (“f composed with g”) is defined by

The domain of consists of the numbers x in the domain of g for which g(x) lies in the domain of f.


Inverse Functions


One-to-One Functions

A function f(x) is one-to-one if every two distinct values for x in the domain, , correspond to two distinct values of the function, .

Properties of a one-to-one function

and

NOTE: A function is one-to-one if and only if its graph intersects each horizontal line at most once.

Inverse Functions

Finding the Inverse of a Function

Step 1: Verify that f(x) is a one-to-one function.Step 2: Let y = f(x).Step 3: Interchange x and y.Step 4: Solve for y.Step 5: Let .Step 6: Note any domain restrictions on .

NOTE: Domain of f = Range of Range of f = Domain of

Example:

1. Find the inverse of the function .2. Find the inverse of the function .3. Find the inverse of the function .

4. Find the inverse of the function .

Even Function, Odd Function

A function is an


even function of x if ,odd function of x if ,

for every x in the function’s domain.


Exponential functions

Function of the form

Even Function (Symmetric about the y-axis)

Odd Function(Symmetric about the origin)


, where a is positive constant is the general exponential function with base a and x as exponent.

The most commonly used exponential function, commonly called natural exponential function is

or

where the base e is the exponential constant whose value is

Rules for exponential functions

i. ~ Product Rule

ii. ~ Quotient Rule

iii. ~ Power Rule

iv. or ~ Reciprocal Rule

v.

Example:Solve the following exponent equations.

(a) (b) 10832 1xx (c)

Logarithmic Functions

The logarithm function with base a, , is the inverse of the base a exponential function .


The function is called the natural logarithm function, and is often called the common logarithm function. For natural logarithm,

Algebraic properties of the natural logarithm

For any numbers and , the natural logarithm satisfies the following rules:

1. Product Rule:

2. Quotient Rule:

3. Power Rule:

4. Reciprocal Rule:

Inverse Properties for and

1. Base a: , ,2. Base e: , ,

Change Base Formula

Every logarithmic function is a constant multiple of the natural logarithm.

NOTE: .

Example 1: Rewrite the following expression in terms of logarithm.

i. ii.

iii. iv.

Example 2:

i. Evaluate .

ii. Simplify .


iii. Simplify

iv. Expand

Example 3: Solve the following equations:

1. 2.

3. 4.

5.

6.

Trigonometric Functions


Hyperbolic Functions

Hyperbolic functions are formed by taking combinations of the two exponential functions and .

The six basic hyperbolic functions

1. Hyperbolic sine of x:

2. Hyperbolic cosine of x:

3. Hyperbolic tangent of x:

4. Hyperbolic cotangent of x:

5. Hyperbolic secant of x:

6. Hyperbolic cosecant of x:

Shifting a Graph of a Function



PROBLET SET: CHAPTER 1

1. Let and find each of the following:

(a) (b) (c) (d) (e)

(f) (g) (h) (i) (j) +

(k) (l)

2. Let . Evaluate and simplify the difference quotient:

3. Find the domain and range of the function defined by each equation.(a) (e)

(b) (f)

(c) (g)

(d) (h)

4. Find and .


(a) (e)

(b) (f)

(c) (g)

(d) (h)

5. Find the inverse g of the given function f, and state the domain and range of g.

(a). (e).(b). (f).

(c). (i).

(d). (j).

6. Determine whether the following functions are odd, even or neither even nor odd.

(a). (f).

(b). (g).

(c). (h).

(d). (i).

(e). (j).

ANSWERS FOR PROBLEM SET: CHAPTER 1

1. (a) (b) (c) (d) (e)

(f) (g) (h) (i) (j)

(k) (l)

2.

3. (a) (e)


(b) (f) (c) (g) (d) (h)

4. (a)(b)

(c)

(d)

(e)(f)(g)(h)

5. (a).(b).

(c).

(d).(e).(f).

6. (a). Neither even nor odd (f) Even(b) Neither even nor odd (g) Odd(c) Neither even nor odd (h) Neither even nor odd(d) Odd (i) Even(e) Odd (j) Odd

Chapter 1 (functions).

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Transcript of Chapter 1 (functions).