Types of Functions

One – One or injective function

A function f: X $\to$Y is said to be one – one or injective if each element in the domain of a function has a distinct image in the co – domain. Example: f:R$\to$R f(x) = 2x is one – one.

Many – one function

A function f: X $\to$Y is said to be many one if there are at least two elements in the domain whose images are the same.

Example: f: R$\to$R given by f(x) = x² is Many – one.

METHODS TO DETERMINE ONE – ONE AND MANY – ONE

$\bullet$ If f(x₁) = f (x₂)$\Rightarrow$x₁ = x₂ for every x₁,x₂ in the domain, then 'f' is one – one else many – one.

$\bullet$ If the function is entirely increasing or decreasing in the domain, then 'f' is one – one else many – one.

Graphical Method

If we draw a line parallel to the x – axis intersect the graph of y = f(x) at one and only one point, then f(x) is one – one and if the line parallel to the x – axis cuts the graph at more than one different points then f(x) is a many – one function.

$\bullet$ Any continuous function f(x) which has at least one local maxima or local minima is many – one.

$\bullet$ All even functions are many one.

$\bullet$ All polynomials of even degree defined on R have at least one local maximum or minima and hence are many one on the domain R. Polynomials of odd degree can be one – one or many – one.

Onto function or Surjective function

A function f: X$\to$Y is said to be a onto function or Surjective function if and only if each element of Y is the image of some element of X i. e. if and only if for every y $\in$ Y there exists some x $\in$ X such that y = f(x). Thus 'f' is onto if f(x) = Y i. e. range = co – domain of function.

Example: The map f: R $\to$[ –1,1] given by f(x) = sin x is an onto map.

Into function

A function f: X$\to$Y is said to be an into function if there exists at least one element in the co – domain Y which is not an image of any element in the domain X. Example: The map f: R $\to$R given by f(x) = x² is an into map

One – one onto map or bijective function

A function f: X $\to$Y is said to be one – one onto or bijective function if and only if

(i) distinct elements of X have distinct images in Y

(ii) each element of Y has at least one pre – image in X.

Example: The map f: X $\to$Y given by f(x) = 2x is a one – one onto map.

Illustration 1 If f: R$\to$R where $f(x)=\frac{x+1}{x^2+x+1}$, find whether f(x) is one – one or many one.

Key concept : If f(x) is a rational function then f(x₁) = f(x₂) will always be satisfied when x₁ = x₂ in the domain.

Hence we can write $f(x_1)-f(x_2)=(x_1-x_2)g(x_1,x_2)=0$ where $g(x_1,x_2)$ is some function in x₁ and x₂. Now if $g(x_1,x_2)$ = 0 gives some solution which is different from x₁ = x₂ and which lies in the domain, then f is many – one else one – one.

Solution: $f(x_1)=f(x_2)\Rightarrow \frac{x_1+1}{{x_1}^2+x_1+1}=\frac{x_2+1}{{x_2}^2+x_2+1}$

One solution of this is obviously x₁ = x₂. Also we have got a relation in x₁ and x₂ and for each value of x₁ in the domain we get a corresponding value of x₂ which may or may not be same as x₁.

If x₁ = 1 we get $x_2=\frac{-1}{2}\ne x_1$, and both lies in the domain of f.

Hence we have two different values x₁ and x₂ for which f(x) has the same value.

Hence ${}^{\prime }{f}^{\prime }$ is many one.