Onto functions
Onto Functions
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An onto function is a type of function that has a specific characteristic in terms of its range and codomain.
Definition
A function $f: A \rightarrow B$ is called an onto function or a surjective function if every element in the codomain $B$ has at least one pre-image in the domain $A$. In other words, for every $b \in B$, there exists at least one $a \in A$ such that $f(a) = b$.
Mathematical Representation
The formal definition of an onto function is:
$$\forall b \in B, \exists a \in A : f(a) = b$$
This means that the range of $f$ is equal to its codomain. The range is the set of all actual outputs of the function, while the codomain is the set that contains the range and possibly more elements that are not outputs of the function.
Examples
Here are some examples to illustrate onto functions:
Onto Function: Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $f(x) = 2x + 3$. This function is onto because for every real number $y$, we can find a real number $x$ such that $f(x) = y$. For example, if $y = 5$, then $x = 1$ satisfies $f(x) = 5$.
Not Onto Function: Let $g: \mathbb{R} \rightarrow \mathbb{R}$ be defined by $g(x) = x^2$. This function is not onto because there is no real number $x$ for which $g(x) = -1$, since squares of real numbers are always non-negative.
Table of Differences and Important Points
| Property | Onto Function (Surjective) | Not Onto Function |
|---|---|---|
| Definition | Every element of the codomain is mapped to by at least one element of the domain. | There is at least one element in the codomain that is not an image of any element of the domain. |
| Range vs Codomain | Range is equal to the codomain. | Range is a proper subset of the codomain. |
| Example | $f(x) = 2x + 3$ for all $x \in \mathbb{R}$ | $g(x) = x^2$ for all $x \in \mathbb{R}$ |
| Mathematical Representation | $\forall b \in B, \exists a \in A : f(a) = b$ | $\exists b \in B, \forall a \in A : f(a) \neq b$ |
Checking if a Function is Onto
To check whether a function is onto, we can use the following steps:
- Identify the codomain of the function.
- Show that for every element in the codomain, there is a pre-image in the domain.
- If you can find at least one element in the codomain without a pre-image, the function is not onto.
Practice Problems
- Determine if the function $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) = 3x - 4$ is onto.
- Is the function $g: \mathbb{Z} \rightarrow \mathbb{Z}$ defined by $g(n) = n^3$ onto?
- Consider the function $h: \mathbb{R} \rightarrow \mathbb{R}^+$ (positive reals) defined by $h(x) = e^x$. Is $h$ onto?
Solutions
- The function $f(x) = 3x - 4$ is onto because for every real number $y$, we can solve the equation $3x - 4 = y$ to find a real number $x$ that maps to $y$.
- The function $g(n) = n^3$ is onto because every integer $m$ has a cube root that is also an integer, which means for every $m \in \mathbb{Z}$, there exists an $n \in \mathbb{Z}$ such that $g(n) = m$.
- The function $h(x) = e^x$ is onto because the exponential function covers all positive real numbers as $x$ ranges over the real numbers.
Understanding onto functions is crucial for various fields of mathematics, including algebra, calculus, and more advanced topics like topology and functional analysis. It is a fundamental concept that helps in classifying functions and understanding their behavior.