# 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

1. Let x1, x2 domain of f and if x1x2 f(x1) ≠ f(x2) for every x1, x2 in the domain, then f is one-one else many-one.
2. Conversely if f(x1) = f(x2) x1 = x2 for every x1, x2 in the domain, then f is one-one else many-one.
3. If the function is entirely increasing or decreasing in the domain, then f is one-one else many-one.
4. Any continuous function f(x) which has at least one local maxima or local minima is many-one.
5. All even functions are many one.
6. 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.
7. If f is a rational function then f(x1) = f(x2) will always be satisfied when x1 = x2 in the domain. Hence we can write f(x1) – f(x2) = (x1x2) g(x1, x2) where g(x1, x2) is some function in x1 and x2. Now, if g(x1, x2) = 0 gives some solution which is different from x1 = x2 and which lies in the domain, then f is many-one else one-one.
8. 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

1. If range = co-domain, then f is onto. If range is a proper subset of co-domain, then f is into.
2. 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 Df.

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 Df.

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) =

Let 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 ∈ Df.