#### Question

Check the injectivity and surjectivity of the following functions: *f*: **Z** → **Z** given by *f*(*x*) = *x*^{3}

#### Solution

*f*: **Z **→ **Z** is given by,

*f*(*x*) = *x*^{3}

It is seen that for *x*, *y* ∈ **Z**, *f*(*x*) = *f*(*y*) ⇒ *x*^{3} = *y*^{3} ⇒ *x* = *y*.

∴ *f* is injective.

Now, 2 ∈ **Z**. But, there does not exist any element *x* in domain **Z** such that *f*(*x*) = *x*^{3} = 2.

∴ *f* is not surjective.

Hence, function *f* is injective but not surjective.

Is there an error in this question or solution?

Solution Check the Injectivity and Surjectivity of the Following Functions: F: Z → Z Given by F(X) = X3 Concept: Types of Functions.