# How do we identify if a function is one-one or many-one and into or onto ??The function f(x) R --> R defined by , 2^(|x|) - 2^(-x) is (a) one-one into (b) one-one onto (c) many-one into (d)...

How do we identify if a function is one-one or many-one and into or onto ??

The function f(x) R --> R defined by ,

2^(|x|) - 2^(-x) is

(a) one-one into

(b) one-one onto

(c) many-one into

(d) many-one onto

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### 1 Answer

The terminology is not completely standard, so here is what I am using:

one-to-one: a function is 1-1 if for every a,b in the domain f(a)=f(b) implies a=b. (injective)

many-one: a function that is not 1-1. (Some list many-one as multivalued functions, which are not true functions)

onto: Every element in the range has an element from the domain mapped to it (Every image has a preimage)(surjective)

into: Every element in the domain gets mapped to an element in the range (the range is a codomain). Note that onto functions are into, but into need not be onto.

Given `f(x)=2^(|x|)-2^(-x)` :

For `x<=0` we have `2^(|x|)=2^(-x)` so for `x<=0,f(x)=0` . This function is not onto, as `f(x)>=0 forall x` so f(x) never maps to a negative number, but it is into as for any x f(x) is real.

For x>0, `2^(|x|)>2^(-x)` . Thus the function is increasing on x>0, and is 1-1. But on x<0, the function is constant and thus not 1-1, so the function is into.

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**This function is into and many-one, so the answer is (c).**

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