Functions and Graphs Mathematics Section UniKL MFI Norhayati Bakri
Definition of a Function Function as a Machine Function defined by an equation Evaluation of Functions
Domain of Function Graphs of Functions Intercepts The Vertical Line Test Piecewise Functions
Definition of a Function A function is a rule that assigns to each element in set X an element in set Y set Y (rule) codomain set X
f
x
domain
f(x)
a f(a) b
output
c input
f(b)=f(c)
f : x ď‚Ž f ( x) f : input ď‚Ž output
range
Injection Moulding Machine
nylon
polythene
DVD / CD storage
Function as a Machine
3
Function can be displayed in tabular form x
Multiply by 4
Add 5
f(x)
3
17
4
21
5
25
Function can be displayed as ordered pairs { ( 3 , 17 ) , ( 4 , 21 ) , ( 5 , 25 ) }
17
A function must assign to each input a UNIQUE output
3 main components in a Function input
output
relationship
A function relates an input to an output. A
A 2K 3K
B
C
C 4K 5K
D employee
2K 3K
B
Function
D
5K
employee
salary
A
4K salary
A
2K 3K
B
4K 5K
C employee
salary
2K
B C
NOT Function
4K
D employee
salary
RELATION between the employee and their salary in company XY
Function defined by an equation A function can be defined implicitly: 5x 2y 3 0 If solve for y:
2y 5x 3
Now it can be defined explicitly:
y
1 (5 x 3 ) 2
1 f ( x ) (5 x 3 ) 2
‘x’ is considered as the input variable (independent variable) ‘y=f(x)’ is considered as the output variable (dependent variable)
Worked example: Does the following equation define y as a function of x? x 2 y 16
Solve for y 16 y 2 x
xy 2 16
Solve for y 16 y x
x2 y 0
When solved, it’s either
2
x can be any real number except 0
x must be any real number greater than 0
FUNCTION
NO
x 0 or y 0
NO
Evaluation of Function To replace or to substitute the function’s variable with a number or an expression. Consider f (x) 1 x x2
If x 2 If x h 3
f (2) 1 (2) (2)2 1 f (h 3) 1 (h 3) (h 3)2 1 h 3 (h2 6h 9) 1 h 3 h2 6h 9 h2 5h 5
Domain of Function Domain of a function is the largest possible set of inputs into the function.
• Domain of polynomial functions is all the real numbers. • Positive radical functions cannot contain a negative number underneath the radical. Set the radicand greater than or equal to zero and solve for the variable. •
Rational functions cannot have zeros in the denominator. Set the denominator to zero and solve for the variable. Then remove the variable from the domain.
Worked example: Domain of polynomial functions is all the real numbers. What is the domain of f ( x) 4x 2 ?
18
14
10
f(x) is a linear function (polynomial) Df ( , )
6
2 -3
-2
-1
-2
-6
0
1
2
3
4
5
6
Worked example: Rational functions cannot have zeros in the denominator. Set the denominator to zero and solve for the variable. Then remove the variable from the domain. x 1 What is the domain of g( x ) ? 4x 2
2 1.5 1
g(x) is a rational function 4x 2 0 x 21
0.5 0 -4
-3
-2
-1
0
-0.5 -1
Dg ( , 21 ) ( 21 , )
-1.5
1
2
3
4
Worked example: Positive radical functions cannot contain a negative number underneath the radical. Set the radicand greater than or equal to zero and solve for the variable. 5
What is the domain of h( x) 4x 2 ?
4
3
h(x) is a positive radical function 4x 2 0
2
x 21
1
0
Dh [ 21 , )
-1
0
1
2
3
4
5
Worked example: What is the domain of J( x )
x 1 ? 4x 2 1.4
4x 2 0 x 21
1.2
1
0.8
DJ ( 21 , )
0.6
0.4
0.2
0 -1
-0.5
0
0.5
1
1.5
2
Worked example: x 1 What is the domain of K( x ) ? 4x 2
Numerator:
x 1 0
x 1
[1, )
Denominator: 4x 2 0 x 21 DJ [1, 21 ) ( 21 , )
Graphs of Functions For a function defined as y f ( x) , we may view •
the inputs x as elements of a horizontal line in the plane
•
the outputs y as elements of a vertical line in the plane The graph of a function in the Cartesian plane is the set of values (x,f(x))
Graphs of functions y=f(x) can be sketched by creating a table of values (x,y) and plotting points on the Cartesian plane.
Worked example: a(x)
Graph a( x )  x
2.5
2
x
a(x)=| x |
-2
2
-1
1
-0.5
0.5
0
0
1
1
2
2
1.5
1
0.5
0 -3
-2
-1
0
1
2
Graph of absolute value function
3
Worked example: b(x) 3
Graph b( x)  x 3
2
x
b(x)
-8
-2
-1
-1
0
0
1
1
8
2
1
0 -9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
-1
-2
-3
Graph of cube root function
6
7
8
9
Characteristics of some simple functions Constant Function
f(x)
f(x)
f ( x) c x
Linear Function
f(x)
x f(x)
m<0
f ( x) mx c
m>0
x
Quadratic Function
f ( x) ax 2 bx c
f(x)
x
f(x)
a<0 x
These characteristics help you to graph the function faster …
a>0 x
Intercepts • •
the x-intercept is a point where the graph crosses the x-axis the y-intercept is a point where the graph crosses the y-axis c(x) 6
x-intercept (3,0)
5
From the graph of
4
3 2
c( x) x 2x 3 2
1 0 -3
x-intercept (-1,0)
-2
-1
-1 -2 -3 -4 -5
y-intercept (0,-3)
0
1
2
3
4
5
Worked example: Determine the intercepts of f ( x) x 4 16 x2 To determine the y-intercept, compute f(0)
f(x) 250
f (0) (0) 16(0) 0 4
2
The y-intercept is (0,0)
200
To determine the x-intercept, solve f(x)=0
150
100
x 4 16 x 2 0 50
x ( x 4)(x 4) 0 2
The x-intercepts are (-4,0) , (0,0) , (4,0)
0 -6
-5
-4
-3
-2
-1
0 -50
1
2
3
4
5
6
The Vertical Line Test In the definition of a function, it is required for a function to have a UNIQUE output for each input. â&#x20AC;˘
If the vertical line touches the graph at exactly one point at the y value of f(x), then the graph is a graph of a function.
â&#x20AC;˘
If the vertical line touches the graph at more than one point at the y value of f(x), then the graph is NOT a graph of a function.
Worked example: Determine which of the following graph is the graph of a function. g(x)
f(x)
intersection
4.00
40
3.00
30
2.00
20
1.00
10
0.00 0
1
2
3
4
5
6
7
8
9
-1.00
10
intersection
0 -4
-3
-2
-1
0
2
-10
-2.00
-20
-3.00
-30
-4.00
-40
Graph of NOT a function
1
Graph of a function
3
4
Piecewise Functions
A piecewise function is a function that obey one rule in one region and another rule in another region. Also known as split function. 5
Rules
Regions
4 3
2x 1 , x 0 f ( x) 2 , 0x2 x
2 1 0 -2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
-1 -2 -3 -4
A two-piece piecewise function
Worked example: Consider the following piecewise function and determine f(0), f(1) and f(3). 2x 2 , x 1 f ( x) 2 x , 1 x 3
When x<1, the function obeys f ( x) 2x 2 f (0) 2(0) 2 2
When 1≤x<3, the function obeys f ( x) 2 x f (1) 2 (1) 1 f (3) undefined
Worked example: Draw the following piecewise function Rules
x 2 f (x) 1 1 x2 2
Regions
, 4 x 2 , 2 x 0 , x0 , 0x3
f ( x) 1 f ( x ) x
x 2 f (x) 1 1 x2 2
f ( x) 2
f ( x ) 21 x 2
, 4 x 2 , 2 x 0 , x0 , 0x3
4 x 2
2 x 0 x0
0x3
Swim with the current Be a good navigator Stay calm under pressure
Thank You See you next week