Question

Page Date which of the the following function are, injection surjection oř bijection on R ?
(a)f(x)= -2x
(b)g(x)= x2+3​

Answer 1

Answer:

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f:R→R, defined by f(x)=x

3

−x

Injectiontest:

Let x

1

and x

2

be any two elements in the domain (R), such that,

f(x

1

)=f(x

2

)

⇒ x

1

3

−x

1

=x

2

3

−x

2

Here, we cannot say x

1

=x

2

.

For example, x

1

=1 and x

2

=−1

⇒ x

1

3

−x

1

=1−1=0

⇒ x

2

3

−x

2

=(−1)

3

−(−1)=−1+1=0

So, 1 and −1 have the same image 0.

So, f is not an injection.

Surjectiontest:

Let y be any element in the co-domain (R), such that f(x)=y for some element x∈R ( Domain ).

⇒ f(x)=y

⇒ x

3

−x=y

By observation we can say that there exist some x in R, such that x

3

−x=y.

So, f is a surjection.

Bijectivetest:

Since, f is surjective but in an injective, then it is not bijective.