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FUNCTIONS - Coggle Diagram
FUNCTIONS
What Is Function?
A function is a correspondence, or rule, that pairs each element of a set(the domain) with exactly one element of another set (the range).
A function relates an input to an output.It is like a machine that has an input and an output. And the output is related somehow to the input.
"f(x) = ... " is the classic way of writing a function.
There are always three main parts:
The input
The relationship
The output
The set of all x's is a subset of set B (Codomain) and is called an image (range).
WHAT IS RULES?
A function is a rule that assigns to each element in a set called
domain
. one and only element in a set called the
codomain
.
A function has to satisfy the following;
1. Every element in the domain is related with an element in the range.
2. Any element in the domain must be related to only one element in the rage.
EXAMPLES
:warning:
Not a function!
It is a function!
Which of the following define a function from A={1, 2, 3, 4} to B={1, 2, 3, 4, 5, 6}
A) f={(1,2), (2,5), (3,1), (4,2)
B) g={(1,2), (2,3), (2,4), (3,5), (4,5)
C) h={(1,1), (2,3), (4,4)
ANSWER:
A İS CORRECT, B AND C ARE FALSE!!
Small letters f, g, h,... are used to name functions!!!
Types Of Functions
1- Into Functions
Let f: A → B be a function. If f(A) ≠ B, f is said to be an
into
function
2- Onto Functions
In mathematics, a function f from a set X to a set Y is surjective (also known as onto, or a surjection), if for every element y in the codomain Y of f, there is at least one element x in the domain X of f such that f(x) = y.
3- One To One Functions
One to one function or one to one mapping states that each element of one set say Set (A) is mapped with a unique element of another set, say, Set (B), where A and B are two different sets. It is also written as 1-1. In terms of function, it is stated as if f(x) = f(y) implies x = y, then f is one to one.
4- Equal Functions
Two functions are equal if they have the same domain and codomain and their values are the same for all elements of the domain.
EXAMPLE :warning:
A={-1, 4}, B{-1, 14, 16=; f: A→B, f(x)=x²-2, g: A→B, g(x)=3x+2 are given. Determine whether f and g are equal or not.
For x= -1
f(-1)= (-1)² -2=-1
g(-1)=3. (-1) +2= -1
For x= 4,
f(4)= 4²-2= 14
g(4)= 3. 4+ 2= 14
F AND G ARE EQUAL( f=g)
5- Identity Function
Let A be a set. The function I: A→ A defined by I(x)=x for all x ∈ A is called
identity function
.
6- Constant Function
Given f: A→ B. if f maps all the elements of set A to one and only one element of set B, this function ,s called as a
constant function
.
EXAMPLE :warning:
f: R→R, f(x)= (m-2)x+m+1 is a constant function;
a) Find m.
b)Find f(1250).
a) f(x)= (m-2)x+m+1, (m-2)=0,
m=2
b) if m is 2, f(x)=(2-2)x+2+1=3, then f(x)=3 and
f(1250)=3
7- Linear Function
For any real numbers a and b, f:R→R, f(x)=y=ax+b is said to be
linear function
.
8- Even And Odd Functions
If f(-x)= f(x) then f is an even function.
If f(-x)= -f(x) then f is an odd function.
EXAMPLES :warning:
R→R, g(x)= x³- 3x
ODD FUNCTION
R→R, f(x)= x²+ 3
EVEN FUNCTION
9-Piecewise Function
Sometimes a function can't be described by a single equation, and instead we have to describe it using a combination of equations. Such functions are called piecewise functions, and probably the easiest way to describe them is to look at the following example.
EXAMPLE :warning:
What is the value of f(6)+ f(-4)+ f(2)
x³-1 , 1<x<6
f(2) belongs to here and f(2)=
7
3- 4x ,6≤x
f(6) belongs here and f(6)=
-21
x+13 , x ≤1
f(-4) belongs to here and f(-4)=
9
Graphs Of Functions
Problems About Graphs Of Functions
EXAMPLE :warning:
Find the domain and the range of the function. Then find the maximum and minimum values of the function.
Vertical Line Test
The vertical line test is used to determine if a graph of a relationship is a function or not. if you can draw any vertical line that intersects more than one point on the relationship, then it is not a function.
Real Life Situations Modelled By Linear Function
Almost any situation where there is an unknown quantity can be represented by a linear equation, like figuring out income over time, calculating mileage rates, or predicting profit. Many people use linear equations every day, even if they do the calculations in their head without drawing a line graph.
Graphs Of Piecewise Function
A piecewise function is a function which have more than one sub-functions for different sub-intervals(sub-domains) of the function's domain. To graph a piecewice function, we graph the different sub-functions for the different sub-intervals of the function's domain.
Graph Of Linear Functions
EXAMPLE :warning:
Sketch the graph of the function f:R→R, f(x)=x+2.
There are three basic methods of graphing linear functions. The first is by plotting points and then drawing a line through the points. The second is by using the y-intercept and slope. And the third is by using transformations of the identity function \displaystyle f\left(x\right)=xf(x)=x.
What Is Graph Of Function?
The set of the points in the analytic plane corresponding to the elements of a function is the graph of this function. Symbolically, given a function f: A→ B, the set f={(x,y):y=f(x), (x,y)∈ AxB} is the graph of the function
Composition Of Functions
Composition of a function is done by substituting one function into another function. For example, f [g (x)] is the composite function of f (x) and g (x). The composite function f [g (x)] is read as “f of g of x”.
EXAMPLE :warning:
f:R→R, g(x)=2x+1. Find the values of (fog)(x) and (fog)(2).
(f (fog)(x)=f(g(x))
f(g(x))=2x+3
f(g(2)=2.2+3=
7
Inverse Of A Function
An inverse function is a function that undoes the action of the another function. A function g is the inverse of a function f if whenever y=f(x) then x=g(y). In other words, applying f and then g is the same thing as doing nothing.
