Methods of finding whether function is One One or Many One
Relation and function of Class 12
Methods of finding whether function is One One or Many One

Methods of finding whether function is One  One or Many  One.
(a) If x_{1} ≠ x_{2} ⇒ f(x_{1}) ≠ f(x_{2}), then f(x) is one  one.
(b) If f (x_{1}) = f (x_{2}) ⇒ x_{1} = x_{2}, and only this then f (x) is one  one.
(c) Any function, which is entirely increasing or decreasing in whole domain, then f (x) is one  one.
(d) Any continuous function f(x), which has at least one local maximum or local minimum, is
many  one.
(e) If any line parallel to xaxis cuts the graph of the function at most at one point, then the function is oneone and if there exists a line which is parallel to xaxis and cuts the graph of the function in at least two points, then the function is many  one.
(f) We put f(x_{1}) = f(x_{2}). Since x_{1} = x_{2} always satisfies f(x1) = f(x_{2}) so (x_{1}  x_{2}) will be a factor of
f(x_{1})  f(x_{2}). Hence we can write as f(x_{1})  f(x_{2}) = (x_{1}  x_{2}) g(x_{1,} x2), where g(x_{1}, x_{2}) is some function of x_{1} and x_{2}. Now if g(x_{1}, x_{2}) = 0 gives only those solutions, which are of the form x_{1} = x_{2}, then f(x) is one  one and if g(x_{1}, x_{2})=0 gives some solution which is different from x_{1} = x_{2}, then f(x) is many  one.
Note: To find whether a function is into or onto, find the range of f(x). If it comes out equal to the codomain, the function is onto otherwise the function is into function.