Question

Classify the following functions according to their type
1.f(x)=x_3

Answer 1

Answer:

f:R→R, defined by f(x)=x3−x

Injectiontest:

Let x1 and x2 be any two elements in the domain (R), such that,

f(x1)=f(x2)

⇒  x13−x1=x23−x2

Here, we cannot say x1=x2.

For example, x1=1 and x2=−1

⇒  x13−x1=1−1=0

⇒  x23−x2=(−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)=