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BMM 104: ENGINEERING MATHEMATICS I Page 1 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
BMM 104: ENGINEERING MATHEMATICS I Page 2 of 13
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
BMM 104: ENGINEERING MATHEMATICS I Page 3 of 13
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.
Example: Attend lecture.
Inverse Functions
BMM 104: ENGINEERING MATHEMATICS I Page 4 of 13
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
BMM 104: ENGINEERING MATHEMATICS I Page 5 of 13
even function of x if ,odd function of x if ,
for every x in the function’s domain.
Example: Attend lecture.
Exponential functions
Function of the form
Even Function (Symmetric about the y-axis)
Odd Function(Symmetric about the origin)
BMM 104: ENGINEERING MATHEMATICS I Page 6 of 13
, 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 .
BMM 104: ENGINEERING MATHEMATICS I Page 7 of 13
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 .
BMM 104: ENGINEERING MATHEMATICS I Page 8 of 13
iii. Simplify
iv. Expand
Example 3: Solve the following equations:
1. 2.
3. 4.
5.
6.
Trigonometric Functions
Example: Attend lecture.
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
BMM 104: ENGINEERING MATHEMATICS I Page 9 of 13
BMM 104: ENGINEERING MATHEMATICS I Page 10 of 13
BMM 104: ENGINEERING MATHEMATICS I Page 11 of 13
Example: Attend lecture.
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 .
BMM 104: ENGINEERING MATHEMATICS I Page 12 of 13
(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)
BMM 104: ENGINEERING MATHEMATICS I Page 13 of 13
(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