Functions

CALCULUS
➢ Calculus is the branch of mathematics

that deals with continuous change.
➢ The meaning of classical calculus is the
study of continuous change of functions.
The two major concepts of calculus are:
• Derivatives
• Integrals
CALCULUS
➢ The derivative is the measure of the rate
of change of a function whereas integral is
the measure of the area under the curve.
➢ The derivative explains the function at a
specific point while the integral
accumulates the discrete values of a
function over a range of values.
FUNCTIONS
Function Notation
How do you
know which
INPUT y= f(x)
variable is
independent
(x)
FUNCTION “f of x”
and which MACHINE
one is
dependent? f
OUTPUT f(x)=y
A function is a rule of correspondence between two nonempty

sets, such that to _____________ in the first set, there
corresponds ____________________
in the second set.
3
In order for a relationship to be a function…
EVERY INPUT MUST HAVE AN OUTPUT
TWO DIFFERENT INPUTS CAN HAVE THE
SAME OUTPUT
ONE INPUT CAN HAVE ONLY ONE
OUTPUT
INPUT
(DOMAIN)
Functions
FUNCTION
MACHINE
OUTPUT (RANGE)
Example 6
Which of the following relations are
functions?
R= {(9,10), (-5, -2), (2, -1), (3, -9)}
S= {(6, a), (8, f), (6, b), (-2, p)}
T= {(z, 7), (y, -5), (r, 7), (z, 0), (k, 0)}
No two ordered pairs can have the

same first coordinate
(and different second coordinates).
Identify the Domain and Range. Then
tell if the relation is a function.
Input Output
-3 3
1 1
3 -2
4
Function?
Yes: each input is mapped
Domain = {-3, 1,3,4} onto exactly one output
Range = {-2,1,3}
Identify the Domain and Range. Then
tell if the relation is a function.
Input Output
-3 3
1 -2
4 1
Domain = {-3, 1,4} Function?

Notice the set notation!!!
Range = {3,-2,1,4} No: input 1 is mapped onto
Both -2 & 1
1. {(2,5) , (3,8) , (4,6) , (7, 20)}
2. {(1,4) , (1,5) , (2,3) , (9, 28)}
3. {(1,0) , (4,0) , (9,0) , (21, 0)}

The Vertical Line Test
If it is possible for a vertical line
to intersect a graph at more
than one point, then the graph
is NOT the graph of a function.
Page 117
Use the vertical line test to visually check if the
relation is a function.
(4,4)
(-3,3)
(1,1)
(1,-2)
Function?
No, Two points are on
The same vertical line.
Use the vertical line test to visually check if the
relation is a function.
(-3,3)
(1,1) (3,1)
(4,-2)
Function?
Yes, no two points are
on the same vertical line
#1 Function?
#2 Function?
#3 Function?
#5 Function?
#6 Function?
#7 Function?
#8 Function?
Function Notation
f (x)
“f of x”
Input = x
Output = f(x) = y
Before… Now…
y = 6 – 3x f(x) = 6 – 3x
x y x f(x)
-2 12 -2 12
-1 9 (x, y) -1 9 (x, f(x))
0 6 0 6
1 3 1 3
2 0 2 0
(input, output)
Function Operations
Addition : ( f + g )( x) = f (x ) + g (x )
Multiplica tion : ( f  g )(x ) = f (x )  g (x )
Subtractio n : ( f − g )(x ) = f (x ) − g (x )
f f (x )
Division :  (x ) = where g(x )  0
g g (x )
Adding and Subtracting Functions
Let f ( x ) = 3 x + 8 and g( x ) = 2 x − 12.

Find f + g and f - g
( f + g )( x) = f (x ) + g (x ) ( f − g )( x) = f (x ) − g (x )
= (3x + 8) + (2 x − 12) = (3x + 8) − (2 x − 12)
= 5x − 4 = x + 20
Multiplying Functions
Let f (x ) = x 2 - 1 and g(x ) = x + 1.

Find f  g
f (x )  g ( x) = ( x 2 − 1)( x + 1)
= x3 + x 2 − x − 1
Dividing Functions
Dividing Functions
Let f (x ) = x 2 - 1 and g(x ) = x + 1.

f 
Find  
g
f (x ) x 2 − 1
= =
g (x ) x + 1
In this case, the domain is
all real numbers EXCEPT
-1, because x=-1 would
( x − 1)( x + 1) give a zero in the
= x −1 denominator.
( x + 1)
Composite Function – When you combine two or
more functions
⚫ The composition of function

g with function is written as
(g  f )(x ) = g ( f (x ))
1
1. Evaluate the inner function f(x) first. 2

2. Then use your answer as the input of
the outer function g(x).
Example 1
Given the functions f (x) = x2 + 6 and g (x) = 2x – 1,

find (f ∘ g) (x).
Solution
Substitute x with 2x – 1 in the function f(x) = x2 + 6.
(f ∘ g) (x) = (2x – 1)2 + 6
= (2x – 1) (2x – 1) + 6
= 4x2 – 4x + 1 + 6
= 4x2 – 4x + 7
Example 2
Given the functions g(x) = 2x – 1 and f(x) = x2 + 6,

find (g ∘ f) (x).
Solution
Substitute x with x2 + 6 in the function g(x) = 2x – 1
(g ∘ f) (x) = 2(x2 + 6) – 1
= 2x2 + 12 – 1
= 2x2 + 11
Evaluate f [g (6)] given that,
f (x) = 5x + 4 and g (x) = x – 3
Solution
First, find the value of f(g(x)).
⟹ f (g (x)) = 5(x – 3) + 4
= 5x – 15 + 4
f (g (x)) = 5x – 11
Now substitute x in f(g(x)) with 6
f (g (6)) = 5(6) – 11
= 30 – 11
= 19
Therefore, f [g (6)] = 19

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CALCULUS

➢ Calculus is the branch of mathematics

A function is a rule of correspondence between two nonempty

No two ordered pairs can have the

Domain = {-3, 1,4} Function?

2. {(1,4) , (1,5) , (2,3) , (9, 28)}

3. {(1,0) , (4,0) , (9,0) , (21, 0)}

Multiplica tion : ( f  g )(x ) = f (x )  g (x )

Let f ( x ) = 3 x + 8 and g( x ) = 2 x − 12.

Let f (x ) = x 2 - 1 and g(x ) = x + 1.

Let f (x ) = x 2 - 1 and g(x ) = x + 1.

⚫ The composition of function

1. Evaluate the inner function f(x) first. 2

Given the functions f (x) = x2 + 6 and g (x) = 2x – 1,

Given the functions g(x) = 2x – 1 and f(x) = x2 + 6,

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