Benginning Calculus Lecture notes 1 - functions

Beginning Calculus
-Real-Valued Functions -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,
FSMT - UPSI
(LECTURE SLIDES SERIES)
VillaRINO DoMath, FSMT-UPSI
(D0) Real-Valued Functions 1 / 18

Functions Increasing and Decreasing Functions New Functions From Old Transformations of Functions Important Functions
Learning Outcomes
Determine the existence of limits of functions
Compute the limits of functions
Determine the continuity of functions.
Connect the idea of limits and continuity of functions.

Functions
De…nition 1
Let X and Y be any two nonempty sets. A function f from X to Y is a
rule that assigns each element x 2 X to a unique element y 2 Y .
f : X ! Y
f (x) = y, x 2 X, y 2 Y
x y
X Y
f

Domain and Range
f (x) = y, x 2 X, y 2 Y
Remark 1
x 2 X is called the independent variable, and y 2 Y is the
dependent variable.
The set X is called the domain of f . The set Y is called the
codomain of f .
The range of f is the set fy 2 Y j y = f (x) , for some x 2 Xg .
The graph of f = f(x, f (x) j x 2 X)g

Monotonic Functions
De…nition 2
Let f be a function on an interval I R
(a) f is called monotonic increasing (nondecreasing) on I if
f (x1) f (x2) x1 < x2; x1, x2 2 I (1)
(b) f is called monotonic decreasing (nonincreasing) on I if
f (x1) f (x2) x1 < x2; x1, x2 2 I (2)

Strictly Increasing and Decreasing Functions
De…nition 3
Let f be a function on an interval I R.
(a) f is called strictly increasing on I if
f (x1) < f (x2) x1 < x2; x1, x2 2 I (3)
(b) f is called strictly decreasing on I if
f (x1) > f (x2) x1 < x2; x1, x2 2 I (4)
Example
y
x-1-5 2 5
D
ecreasing
Increasing
Constant

Example
Each of the functions f , g, h in the table below is increasing, but each
increases in a di¤erent way. Match each function with its possible graph.
x f (x) g (x) h (t)
1 23 10 2.2
2 24 20 2.5
3 26 20 2.8
4 29 37 3.1
5 33 44 3.4
6 38 50 3.7
(a) (b) (c)

Algebra of Functions
De…nition 4
Given two functions f and g. De…ne f + g, f g, f g, and
f
g
as follows
(a) (f + g) (x) = f (x) + g (x) .
(b) (f g) (x) = f (x) g (x) .
(c) (f g) (x) = f (x) g (x) .
(d)
f
g
(x) =
f (x)
g (x)
, g (x) 6= 0.

Composition of Functions
De…nition 5
Let f : X ! Y 0 and g : Y ! Z be functions. De…ne a new function
g f : X ! Z as follows:
(g f ) (x) = g (f (x)) , 8x 2 X (5)
The function g f is called the composition of f and g.

Inverse of Functions
De…nition 6
Let f : X ! Y be a one-to-one correspondence function. Then f has
an inverse f 1 and,
f 1
(y) = x , y = f (x) x 2 X, y 2 Y (6)

Even and Odd Functions
De…nition 7
For any function f ,
(a) f is called an even function if f ( x) = f (x) , for all x in the
domain.
(b) f is called an odd function if f ( x) = f (x) , for all x in the
domain.

Example
f (x) = x2 and g (x) = ex2
are examples of even functions; h (x) = x3
and l (x) = x1/3 are examples of odd functions.
y
x
(-x, y) (x, y)
y
x
(-x, -y)
(x, y)
y
x
(x, -y)
(x, y)

Re‡ecting
De…nition 8
Let y = f (x). Then,
(a) the re‡ection of the graph of y = f (x) in the x axis is the graph
of y = f (x) .
(b) the re‡ection of the graph of y = f (x) in the y axis is the graph
of y = f ( x) .

Example
The re‡ection of f (x) = x2 in the x axis is the graph of
f (x) = x2.
-4 -2 2 4
-20
-10
10
20
x
y f(x)
g(x) = -f(x)

Example
The re‡ection of f (x) = x2 in the y axis is the graph of
f ( x) = ( x)2
.
-4 -2 2 4
-20
-10
10
20
x
y g(x) = f(-x)
f(x)

Shifting
De…nition 9
Let c > 0 is any constant. Vertical and horizontal shifts in the graph of
y = f (x) are represented as follows:
(a) Vertical shift c units upward: g (x) = f (x) + c;
(b) Vertical shift c units downward: g (x) = f (x) c;
(c) Horizontal shift c units to the right: g (x) = f (x c)
(d) Horizontal shift c units to the left: g (x) = f (x + c)

Example
Let f (x) = x3. Then the graph of a function:
g (x) = x3 + 3 = f (x) + 3 is obtained by shifting the graph of
f (x) = x3, 3 unit upward.
-4 -2 2 4
-4
-2
2
4
x
y
f(x)
g(x) = f(x)+3
g (x) = x3 3 = f (x) 3 is obtained by shifting the graph of
f (x) = x3, 3 unit downward.
4y f(x)VillaRINO DoMath, FSMT-UPSI

Example - continue
g (x) = (x + 3)3
= f (x + 3) is obtained by shifting the graph of
f (x) = x3, 3 unit to the left..
-4 -2 2 4
-4
-2
2
4
x
y
f(x)
g(x) = f(x+3)
g (x) = (x 3)3
= f (x 3) is obtained by shifting the graph of
f (x) = x3, 3 unit to the right..
4y
f(x)VillaRINO DoMath, FSMT-UPSI

Benginning Calculus Lecture notes 1 - functions

More Related Content

What's hot (20)

Similar to Benginning Calculus Lecture notes 1 - functions (20)

Recently uploaded (20)

Benginning Calculus Lecture notes 1 - functions