Functions Class 11

 Functions 

Function

● Let  `X` and `Y` be two nonempty sets. Then, a relation `f` from `X` to `Y` is called a function, if every element in `X` has a unique image in `Y`, and we write, `f: X \rightarrow Y`.

Thus, a relation `f` from `X` to `Y` is a function, if dom `(f) = X` and no two distinct ordered pairs in `f` have the same first coordinate.  

If `(x,y) \in f`, we write `f(x) = y`

Here, `y` is called the image of `x` under `f` and `x` is called the pre-image of `y`. 

If `f: X \rightarrow Y` then dom`(f) = X` and range `(f) \subseteq Y`.

Also, `Y` is called the co-domain of `f`.

● dom `(f) = {x: (x,y) \in f} = X`

● range `(f) = {y: (x,y) \in f} \subseteq Y` 

● co-domain `(f) = Y`

Real Function

● A function which has either `R` or one of its subsets as its range is called a real-valued function. Further, if its domain is also either `\R` or a subset of `R`, it is called a real function. 

Types of Functions

1. Identity function
2. Constant function
3. Polynomial function
4. Rational function
5. Modulus function
6. Signum function
7.  Greatest integer function 

1. Identity function

The function `f : R → R` defined by `y = f (x) = x,  forall x ∈ R` is called the identity function. 

The graph is a straight line as shown in Fig.  It passes through the origin.

Dom `(f )= R` 

Range `(f) = R`

2. Constant function

The function `f : R → R` defined by `y = f (x) = c, forallx ∈ R`, where `c` is a constant `∈ R`, is a constant function. 

Dom `(f) = R` 

Range `(f )= {C}`

3. Polynomial function 

A function `f : R → R` is said to be polynomial function if `forall x in R,  y = f (x) = a_0 + a_1 x + a_2 x ^2 +  a_n x^ n` , where `n` is a non-negative integer and  `a_0 , a_1 , a_2 ,...,a_n∈R`.

4. Rational functions 

Rational functions are functions of the type `\frac{f(x)}{g(x)}`, where `f(x)` and `g(x)` are polynomial functions of `x` defined in a domain, where `g(x) ≠ 0`.

 5. Modulus function 

The function `f: R→R` defined by `f(x) = |x|  forall x ∈R` is called modulus function. 

`|x| ≥ 0  forall x \in R`

`therefore` dom`(f) = R`

range `(f) = [0, ∞)`

Algebra of real functions

(i)  Addition of two real functions

(ii) Subtraction of a real function from another

(iii) Multiplication by a scalar

(iv) Multiplication of two real functions

(v) Quotient of two real functions 

(i)  Addition of two real functions

Let `f : X → R` and `g : X → R` be any two real functions, where `X ⊂ R`. 

Then, we define `(f + g): X → R` by `(f + g) (x) = f (x) + g (x), \forall x ∈ X`.

(ii) Subtraction of a real function from another 

Let `f : X → R` and `g: X → R` be any two real functions, where `X ⊂ R`. 

Then, we define `(f – g) : X→R` by `(f–g) (x) = f(x) –g(x), \forall x ∈ X`. 

References:

1. NCERT Mathematics Class 11
2. NCERT Exemplar Class 11