FUNCTIONS -

Functions Theorems Question number 18:

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TYPES OF FUNCTIONS

1. A function f : A→B is said to be one one function or injection from A into B if different elements in A have different f-images in B.

2. A function f: A→ B is one one if x₁, x₂ ∈A, x₁ ≠ x₂ ⇔ f (x₁) ≠ f (x₂)

3. A function f: A →B is one one iff x1, x2 ∈ A, f (x1) = f (x2)⇔ _x1 = x₂

4. If A, B are two finite sets, then the number of one one functions that can be defined from A into B is n(B)^Pn(A)

5. A function f : A →B is said to be onto function or surjection from A onto B if f(A)=B.

6. A function f : A →B is onto iff y∈ B, x∈ A, f(x)=y.

7. If A, B are two finite sets and n(B) = 2, then the number of onto functions that can be defined from A onto B is 2^n(A) - 2.

8. A function f : A→B is said to be one one onto function or bijection from A onto B if f : A→ B is both one one function and onto function.

9. A function f: A→ B is one one onto iff

10. If A, B are two finite sets and n(A) = n(B), then the number of bijections that can be defined from A onto B is n(A)!.

11. If f : A →B, g : B→ C are two functions then the function gof: A →C defined by (gof)(x) = g[f(x)] x A is called composite function of f and g.

12. If f : A →B, g : B →C are two one one function then gof: A →C is also one one.

13. If f : A→ B, g: B→ C are two onto functions, then gof: A →C is also onto.

14. If f : A→ B, g : B →C are two one one onto functions then gof: A →C is also one one onto.

15. Two functions f : A →B, g : C→ D are said to be equal if 1) A = C ii) f (x) = g(x) x A. It is denoted by f = g.

16. If f: A→ B, g : B →C, h : C→ D are three functions, then ho(gof) = (hog)of.

17. If A is a set, then the function I on A defined by I(x) =x x A, is called identify function on A. It is denoted by IA.

18. If f: A B and IA, lB are identify functions on A, B respectively then foIA = IBof = f.

19. If f: A A then foI = Iof = f, where I is the identity function on A.

21. Finite Set: If A is empty or there exists nN such that there is a bijection from A onto {1,2,3,...,n} then A is called a finite Set.
In such a case we say that number of elements in A is n and denote it |A| or n(A).
22. Equality of a fuction: Let f and g be fuctions. We say f and g are Equal and write f = g
if domain of f = domain of g and f(x) = g(x) for all x domain f.
23. Constant fucntion: A fucntion f: A B is said be a Constant fucntion, if the range of f contains one and only one element
i.e. f(x) = c for all xA, for some fixed cB. In this case the constant function f will be denoted by 'c' itself.

Function – Definition

An algebraic function is a type of equation that uses mathematical operations. An equation is a function if there is a one-to-one relationship between its x-values and y-values.

Algebraic Functions

An algebraic function is a function that involves only algebraic operations, like, addition, subtraction, multiplication, and division, as well as fractional or rational exponents. Think of an algebraic function as a machine, where real numbers go in, mathematical operations occur, and other numbers come out.

We call the numbers going into an algebraic function the input, x, or the domain. Any number can go into a function as long as it is not divided by zero or does not produce a negative square root. A function can preform many mathematical operations with a domain as long as the range is one value for each domain used. We call the numbers coming out of a function the output, y, or the range. Remember, one value in, one value out.

There are many different types of algebraic functions: linear, quadratic, cubic, polynomial, rational, and radical equations. In this next part of the lesson, we’ll learn about a couple of different methods we can use to identify them.

Function Machine

Tables

One way of identifying an algebraic function is through the use of a table, which can show us if there is one domain and one range. Sometimes functions add to the domain to get the range, like x + 2. Sometimes functions multiply the domain to get the range, like 3x. Functions may also subtract or divide the domain or use a combination of operations to produce the range. As long as the rule of ‘one in/one out’ is kept in place, the function exists.

If an algebraic function says to add two to the domain, we can create a table to show the function:

As you can see, for every domain, we have one range. These pairs of x values- and y-values are called ordered pairs because we put them in order (x,y).

We can also turn our table into ordered pairs to show a function: (1,3), (4,6), (-2,0) and (-3,-1) where there is one x-value for every one y-value.

Graphs

We can also use graphs to identify functions by plotting ordered pairs onto a Cartesian Coordinate System, where the x-values are on the horizontal line and the y-values are on the vertical line. Where the ordered pairs meet is where the point is graphed. If we plot the points, we end up with a straight line, so the function, x + 2, is considered a linear function and can be written in functional notation as f(x) = x + 2. The f(x) is just another way to write y, which we call the f-function. It is a way for us to identify the different functions, instead of calling them all y = …

Linear Function

Vertical Line Test

We know a graph is a function if it can pass the vertical line test. In this test, if we place a vertical line anywhere on a graph, it will cross in only one place. If a vertical line crosses in two places on a graph, it is in conflict with the one in, one out rule. So, it is not a function.

Vertical Line Test on Linear Function

Here is an example of a graph that is not a function.

Not a Function

Examples of Functions

As we said at the beginning of the lesson, there are many types of functions, such as the quadratic function and the cubic function. Let’s start with a quadratic function.

The quadratic function: g(x) = x^2 – 3. First, let’s create a table.

The domain can be any real number, this is why the x-value or domain is called the independent variable. Here we’ll use -2, -1, 0, 1, and 2 for the domain.

The range or y-value is called the dependent variable because it depends on what we use for the x-term.

A function or mapping (Defined as $f : X \to Y$ ) is a relationship from elements of one set X to elements of another set Y (X and Y are non-empty sets). X is called Domain and Y is called Codomain of function ‘f’.

Function ‘f’ is a relation on X and Y such that for each $x \in X$ there exists a unique $y \in Y$ such that $(x, y) \in R$ . ‘x’ is called pre-image and ‘y’ is called image of function f.

A function can be one to one or many to one but not one to many.

Injective / One-to-one function

A function $f : A \to B$ is injective or one-to-one function if for every $b \in B$ , there exists at most one $a \in A$ such that $f (s) = t$

This means a function f is injective if $a_{1} \neq a_{2}$ implies $f (a 1) \neq f (a 2)$ .

Example
- $f : N \to N, f (x) = 5 x$ is injective.
- $f : N \to N, f (x) = x^{2}$ is injective.
- $f : R \to R, f (x) = x^{2}$ is not injective as $(- x)^{2} = x^{2}$
Surjective / Onto function

A function $f : A \to B$ is surjective (onto) if the image of f equals its range. Equivalently, for every $b \in B$ , there exists some $a \in A$ such that $f (a) = b$ . This means that for any y in B, there exists some x in A such that $y = f (x)$

Example
- $f : N \to N, f (x) = x + 2$ is surjective.
- $f : R \to R, f (x) = x^{2}$ is not surjective since we cannot find a real number whose square is negative.
Bijective / One-to-one Correspondent

A function $f : A \to B$ is bijective or one-to-one correspondent if and only if f is both injective and surjective.

Problem

Prove that a function $f : R \to R$ defined by $f (x) = 2 x - 3$ is a bijective function.

Explanation − We have to prove this function is both injective and surjective.

If F $f (x_{1}) = f (x_{2})$ , then $2 x_{1} - 3 = 2 x_{2} - 3$ and it implies that $x_{1} = x_{2}$ Hence, f is injective.

Here, $2 x - 3 = y$
$2 x - 3 = y$ So, $x = (y + 5) / 3$ which belongs to R and $f (x) = y$ .Hence, f is surjective.Since f is both surjective and injective, we can say f is bijective.

Inverse of a Function

The inverse of a one-to-one corresponding function $f : A \to B$ is the function $g : B \to A$ , holding the following property −

$f (x) = y \Leftrightarrow g (y) = x$
The function f is called invertible, if its inverse function g exists.

Example
- A Function $f : Z \to Z, f (x) = x + 5$ , is invertible since it has the inverse function $g : Z \to Z, g (x) = x - 5$
- A Function $f : Z \to Z, f (x) = x^{2}$ is not invertiable since this is not one-to-one as $(- x)^{2} = x^{2}$ .
Composition of Functions

Two functions $f : A \to B$ and $g : B \to C$ can be composed to give a composition $g o f$ . This is a function from A to C defined by $(g o f) (x) = g (f (x))$

Some Facts about Composition
- If f and g are one-to-one then the function $(g o f)$ is also one-to-one.
- If f and g are onto then the function $(g o f)$ is also onto.
- Composition always holds associative property but does not hold commutative property.