# One-one and many-one functions

If each element in the domain of a function has a distinct image in the co-domain, the function is said to be one-one. One-one functions are also called

**injective**functions.# Methods to determine one-one and many-one

- Let
*x*_{1},*x*_{2}∈ domain of*f*and if*x*_{1}≠*x*_{2}⇒*f*(*x*_{1}) ≠*f*(*x*_{2}) for every*x*_{1},*x*_{2}in the domain, then*f*is one-one else many-one. - Conversely if
*f*(*x*_{1}) =*f*(*x*_{2}) ⇒*x*_{1}=*x*_{2}for every*x*_{1},*x*_{2}in the domain, then*f*is one-one else many-one. - If the function is entirely increasing or decreasing in the domain, then
*f*is one-one else many-one. - Any continuous function
*f*(*x*) which has at least one local maxima or local minima is many-one. - All even functions are many one.
- All polynomials of even degree defined in
*R*have at least one local maxima or minima and hence are many one in the domain*R*. Polynomials of odd degree can be one-one or many-one. - If
*f*is a rational function then*f*(*x*_{1}) =*f*(*x*_{2}) will always be satisfied when*x*_{1}=*x*_{2}in the domain. Hence we can write*f*(*x*_{1}) –*f*(*x*_{2}) = (*x*_{1}–*x*_{2})*g*(*x*_{1},*x*_{2}) where*g*(*x*_{1},*x*_{2}) is some function in*x*_{1}and*x*_{2}. Now, if*g*(*x*_{1},*x*_{2}) = 0 gives some solution which is different from*x*_{1 }=*x*_{2}and which lies in the domain, then*f*is many-one else one-one. - Draw the graph of
*y*=*f*(*x*) and determine whether*f*(*x*) is one-one or many-one.

# Onto and into functions

Let

*f*:*X*→*Y*be a function. If each element in the co-domain “*Y*” has at least one pre-image in the domain*X*, that is, for every*y*∈*Y*three exists at least one element*x*∈*X*such that*f*(*x*) =*y*, then*f*is onto. In other words range of*f*=*Y*, for onto functions.On the other hand, if there exists at least one element in the co-domain

*Y*which is not an image of any element in the domain*X*, then*f*is into.Onto function is also called

**surjective****function**and a function which is both one-one and onto is called**bijective****function**.# Methods to determine onto or into

- If range = co-domain, then
*f*is onto. If range is a proper subset of co-domain, then*f*is into. - Solve
*f*(*x*) =*y*for*x*, say*x*=*g*(*y*). Now if*g*(*y*) is defined for each*y*∈ co-domain and*g*(*y*) ∈ domain of*f*for all*y*∈ co-domain, then*f*(*x*) is onto. If this requirement is not met by at least one value of*y*in co-domain, then*f*(*x*) is into.

*Remark:*- An into function can be made onto by redefining the co-domain as the range of the original function.
- Any polynomial function
*f*:*R*→*R*is onto if degree is odd; into if degree of*f*is even.

# Even and odd functions

**Even function**A function

*y*=

*f*(

*x*) is said to be an even functions if

*f*(–

*x*) =

*f*(

*x*) ∀

*x*∈

*D*.

_{f}Graph of an even function

*y*=*f*(*x*) is symmetrical about the*y*-axis, i.e., if point (*x*,*y*) lies on the graph then (–*x*,*y*) also lies on the graph.**Odd function**A function

*y*=

*f*(

*x*) is said to be an odd function if

*f*(–

*x*) = –

*f*(

*x*) ∀

*x*∈

*D*.

_{f}Graph of an odd function

*y*=*f*(*x*) is symmetrical in opposite quadrants. i.e., if point (*x*,*y*) lies on the graph then (–*x*, –*y*) also lies on the graph.

*Notes:*- Sometimes it is easy to prove that
*f*(*x*) –*f*(–*x*) = 0 for even function and*f*(*x*) +*f*(–*x*) = 0 for odd functions. - A function can be even or odd or neither.
- Every function can be expressed as a sum of an even and an odd function i.e.,
*f*(*x*) =*h*(*x*) = and*g*(*x*) = . It can now easily beshown that*h*(*x*) is even and*g*(*x*) is odd. - The first derivative of an even function is an odd function and vice versa.
- If
*x*= 0 ∈ domain of*f*, then for odd function*f*(*x*) which is continuous at*x*= 0,*f*(0) = 0, i.e., if for a function,*f*(0)*≠*0, then that function cannot be odd. It follows that for a differentiable even function*f*′(0) = 0, i.e., if for a differentiable function*f*′(0) ≠ 0 then the function*f*cannot be even. *f*(*x*) = 0 is the only function which is defined on the entire number line is even and odd at the same time.- Every even function
*y*=*f*(*x*) are many-one ∀*x*∈*D*._{f}