Question

A function f: R-R , f(x) = x/(x^2 + 1). check whether the function is one-one function​

Answer 1

Answer:

Given, function f:R→R such that f(x)=1+x  

2

,  

Let A and B be two sets of real numbers.

Let x  

1

​  

,x  

2

​  

∈A such that f(x  

1

​  

)=f(x  

2

​  

).

⇒1+x  

1

2

​  

=1+x  

2

2

​  

⇒x  

1

2

​  

−x  

2

2

​  

=0⇒(x  

1

​  

−x  

2

​  

)(x  

1

​  

+x  

2

​  

)=0

⇒x  

1

​  

=±x  

2

​  

. Thus f(x  

1

​  

)=f(x  

2

​  

) does not imply that x  

1

​  

=x  

2

​  

.

For instance, f(1)=f(−1)=2, i.e. , two elements (1, -1) of A have the same image in B. So, f is many-one function.

Now, y=1+x  

2

⇒x=  

y−1

​  

⇒elements < y have no pre-image in A (for instance an element -2 in the codomain has no pre-image in the domain A). So, f is not onto.

Hence, f is neither one-one onto. So, it is not bijective.

solution