# TOC Functions

TOC Functions is the important topic of the Subject Theory Of Computation.

**Prove that Functions f: R → R, f(x) = x^{2} is not one-to-one and not onto function.**

The range and codomain of f(x) = x^{2} are not equal or every element of codomain is not actually one of the values of the function. So function f is not in function.

The function is not one to one because every element y of B is f(x) for more than one x in A. EX: f(-1) = f(1) =1.

**Prove that Functions f: R → R^{+}, f(x) = x^{2} is not one-to-one and onto function.**

The range and codomain of f(x) = x^{2} are equal or every element of codomain is actually one of the values. Of the function. So function f is onto function.

The function is not one to one because every element y of B is f(x) for more than one x in A. EX: f(-1) = f(1) =1.

**Prove that Functions f: R^{+ }→ R, f(x) = x^{2} is one-to-one and not onto function.**

The range and codomain of f(x) = x^{2} are not equal or every element of codomain is not actually one of the values of the function. So function f is not onto function.

Given function is one to one because no single element of codomain can be f(x) for more than one element in the domain.

#### Prove that *f: R*^{+ }→ R^{+}, *f*(x) = x^{2} is one-to-one and onto function. (bijection).

The range and codomain of f(x) = x^{2} equal or every element of codomain are actually one of the values of the function. So function f is onto function.

Given function one to one because no single element of codomain can be f(x) for more than one element in the domain.

The function *f*(x) = x^{2} is onto function as well as one-to-one function. So, it is called as bijection function.

**Related Terms**

Theory of Computation, Basic Terms of TOC

## Leave a Reply