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 x1 ≠ x2 ⇒ f(x1) ≠ f(x2), then f(x) is one - one.

(b) If f (x1) = f (x2) ⇒ x1 = x2, 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 x-axis cuts the graph of the function at most at one point, then the function is one-one and if there exists a line which is parallel to x-axis and cuts the graph of the function in at least two points, then the function is many - one.

(f) We put f(x1) = f(x2). Since x1 = x2 always satisfies f(x1) = f(x2) so (x1 - x2) will be a factor of
f(x1) - f(x2). Hence we can write as f(x1) - f(x2) = (x1 - x2) g(x1, x2), where g(x1, x2) is some function of x1 and x2. Now if g(x1, x2) = 0 gives only those solutions, which are of the form x1 = x2, then f(x) is one - one and if g(x1, x2)=0 gives some solution which is different from x1 = x2, 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 co-domain, the function is onto otherwise the function is into function.

 

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