Question

if a function is neither one-one nor onto , can we say it's a function? if yes , then which function. like this example f:A to B y=f(x)=x^2 +3 A={-1,1,2,3} B={4,5,6,7}
THIS IS NEITHER ONE-ONE NOR ONTO. SO , if it is a function, so what is the name of this ?

Answer 1

Answer on Question #51699 – Math – Algebra

if a function is neither one-one nor onto, can we say it's a function? if yes, then which function. like this example f:A to B y=f(x)=x^2 + 3 A={-1,1,2,3} B={4,5,6,7}

THIS IS NEITHER ONE-ONE NOR ONTO. SO, if it is a function, so what is the name of this?

Solution

Main parts of function: the input; the relationship, the output.

**Definition 1.** A function $f$ from a set $A$ to a set $B$ relates each element of $A$ with exactly one element of $B$ (the same set $A = B$ is possible).

In other words, every element in $A$ is related to some element in $B$ (But some elements of $B$ might not be related to at all, which is fine).

Besides, a function is single valued, that is, function will not give back two or more results for the same input (for example, "$f(2) = 5$, $f(2) = 6$" is not correct).

The one-to-many case "$f(a_1) = b_1$, $f(a_1) = b_2$" is not allowed, but many-to-one case "$f(a_1) = b_1$, $f(a_2) = b_1$" is allowed.

**Definition 2.** A function $f: A \to B$ is called **onto** if for all $b$ in $B$ there is an $a$ in $A$ such that $f(a) = b$.

Then all elements in $B$ are used.

**Definition 3.** A function $f: A \to B$ is called **one-to-one** if whenever $f(a_1) = f(a_2)$ then $a_1 = a_2$.

No element of $B$ is the image of more than one element in $A$.

The neither one-to-one nor onto function (in other words neither injective nor surjective function) does not have special name. It is a function in general.

Example 1. If $f \colon A \to B$, where $A = \mathbb{R}$, $B = \mathbb{R}$, then $f(x) = x^2$ is an example of neither one-to-one nor onto function. It is not onto, since the image of $f(x)$ is $[0; +\infty)$, and not one-to-one, since $f(-1) = f(1)$.

Example 2. A function in this question (f: $A \to B$, where $A = \{-1,1,2,3\}$, $B = \{4,5,6,7\}$, $y = f(x)$, $f(x) = x^2 + 3$) is not defined at point 3, because a function has only one relationship for each input value from $A$, but $y(3) = 3^2 + 3 = 12$ and 12 is not element of $B$. In other words, element 3 from $A$ is not assigned to any element of $B$.

Example 3. If we take $y = f(x)$, $f(x) = x^2 + 3$, $A = \{-1, 1, 2, 3\}$, $B = \{4, 5, 7, 12\}$, then function $f: A \to B$ is well defined, it is onto function, but not one-to-one (here $f(-1) = f(1)$).

Example 4. By means of the least squares method we can suggest another function $y = f(x)$, for example, $y = 4.25 + 0.541667x + 0.25x^2 - 0.041667x^3$, which approximately represents $f \colon A \to B$, where $A = \{-1, 1, 2, 3\}$, $B = \{4, 5, 6, 7\}$.

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