Question

Let $f:R \to R$ be defined as $f(x) = 3x.$ Choose the correct answer.
(A) $f$ is one-one onto
(B) $f$ is many-one onto
(C) $f$ is one-one but not onto
(D) $f$ is neither one-one nor onto

Answer 1

A function is a relationship that states that each input should have only one output. The relationship between INPUT and OUTPUT is depicted by the graph. A function, on the other hand, is a relation that produces one OUTPUT for each INPUT.

Complete answer:
$f(x) = 3x.$ is the concept of $f:R \to R$ .
Let $x,y,R$ be such that $f(x) = f(y)$
$\Rightarrow 3x = 3y$
$\Rightarrow x = y$
Therefore, $f$ stands for "one-one" relation.
Furthermore, there exists $y/3$ in $R$ such that
$f(y/3) = 3(y/3) = y$ (for any real number y in the co-domain $R$ ).
f stands for “onto” relation.
As a result, function f is one-one and onto. A is the right answer.

Additional Information:
Injective function or one-to-one function: If there is a distinct element of Q for each element of P, the function f: P Q is said to be one to one.
Many to one: Two or more elements of P are mapped to the same element of set Q by this function.
Onto Function/Surjective Function: A function that has a pre-image in set P for each element of set Q.
One-one correspondence or Bijective function: The function $f$ matches with each element of P with a discrete element of Q and every element of Q has a pre-image in P.

Note:
A relation can be of many types such as, Reflexive, symmetric, transitive relation. On the basis of relation mapping can be done. The ordered pair is generated by the object from each set, and the relation between the two sets is known as the collection of the ordered pair.