Definition of a unique function

In summary, the conversation discussed the difference between two functions with the same expression and differing domains. While some people may consider them equal based on a simplified definition of a function, the formal definition includes the domain and codomain as part of the function's identity. Therefore, the two functions with different domains are considered different functions.
  • #1
jonsploder
2
0
Hi all, I'm wondering whether an expression which is used to describe a function in a certain domain is a different function for the same expression with a differing domain.

For example: expression; x^2.
f(x) = x^2 for domain {1 < x < 10}
f(x) = x^2 for domain {10 < x < 11}

Are these two f(x)'s the same function, or different functions, by definition. I couldn't be sure by Wikipedia, and it's a difficult question to type into a search engine.
 
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  • #2
hi jonsploder! welcome to pf! :smile:
jonsploder said:
… Are these two f(x)'s the same function, or different functions …

they're different

they're both restrictions of the same function defined on the whole of R :wink:
 
  • #3
Thanks for the welcome, and the reply.
I know that they are different, however I was wondering, by the most formal definition of a function, whether they are different functions, or if indeed the domain of a function constitutes its identity as a function.
 
  • #4
jonsploder said:
I know that they are different, however I was wondering, by the most formal definition of a function, whether they are different functions, or if indeed the domain of a function constitutes its identity as a function.

they're functions, and they're different

so they're different functions

the definition of a function includes its range and domain: different range and/or domain, different functions
 
  • #5
jonsploder said:
Thanks for the welcome, and the reply.
I know that they are different, however I was wondering, by the most formal definition of a function, whether they are different functions, or if indeed the domain of a function constitutes its identity as a function.

The domain and codomain are part of the definition of a function.

Two functions [itex]f : A \to B[/itex] and [itex]g : C \to D[/itex] are equal if and only if [itex]A = C[/itex] and [itex]B = D[/itex] and for all [itex]a \in A[/itex], [itex]f(a) = g(a)[/itex].
 
  • #6
pasmith said:
Two functions [itex]f : A \to B[/itex] and [itex]g : C \to D[/itex] are equal if and only if [itex]A = C[/itex] and [itex]B = D[/itex] and for all [itex]a \in A[/itex], [itex]f(a) = g(a)[/itex].
This definition is exactly correct. That should be your definition.

...

It's worth noting, however, that sometimes people get lazy about codomains and say [itex]f : A \to B[/itex] and [itex]g : C \to D[/itex] are equal when [itex]A = C[/itex] and for all [itex]a \in A[/itex], [itex]f(a) = g(a) \in B\cap D[/itex].
 
  • #7
economicsnerd said:
It's worth noting, however, that sometimes people get lazy about codomains and say [itex]f : A \to B[/itex] and [itex]g : C \to D[/itex] are equal when [itex]A = C[/itex] and for all [itex]a \in A[/itex], [itex]f(a) = g(a) \in B\cap D[/itex].

To explain why this definition is bad, consider
##f : \mathbb{R} \to \mathbb{R}##, ##f(x) = 0##
##g : \mathbb{R} \to \{0\}##, ##g(x) = 0##.
Note that under the definition economicnerd gave these would be considered equal. However, g is a surjection while f is not.
 

What is a unique function?

A unique function is a mathematical concept that describes a relationship between inputs and outputs, where each input has only one corresponding output. This means that for every input value, there is only one output value, making the function distinct and one-of-a-kind.

How is a unique function different from a regular function?

While all functions have a unique relationship between inputs and outputs, a unique function has the additional property of having only one output for each input. Regular functions may have multiple outputs for the same input, making them non-unique.

What are some examples of unique functions?

Examples of unique functions include linear functions, quadratic functions, and exponential functions. These functions have a distinct and unambiguous relationship between inputs and outputs, with no repeated outputs for any given input.

Why is the concept of a unique function important in mathematics?

Unique functions are essential in mathematics because they allow us to model and describe real-world phenomena, make predictions, and solve problems. They also help us understand the fundamental concepts of functions and their properties.

How can we determine if a function is unique?

To determine if a function is unique, we can use the vertical line test. If a vertical line can intersect a graph of the function at more than one point, then the function is not unique. If the vertical line intersects the graph at only one point, then the function is unique.

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