Determine which set is a function

In summary, a function is a relation between two sets where each element in the first set is paired with exactly one element in the second set. A one-to-one function is a special type of function where each element in the second set is paired with exactly one element in the first set. The vertical line test is used to determine if a relation is a function, while the horizontal line test is used to determine if the function is one-to-one.
  • #1
nycmathguy
Homework Statement
Determine which set is a function.
Relevant Equations
n/a
Here is the fuzzy definition of a function as presented by Ron Larson.

Definition of Function

A function f from a set A to a set B is a relation that assigns to each element x
in the set A exactly one element y in the set B. The set A is the domain (or set
of inputs) of the function f, and the set B contains the range (or set of outputs).

Larson goes on to say:

The ordered pairs below can represent a function. The first coordinate (x-value) is
the input and the second coordinate (y-value) is the output.

{(1, 9), (2, 13), (3, 15), (4, 15), (5, 12), (6, 10)}

Let me see.

The above set of elements is a function because every x-value is matched to a unique y-value. Correct?

I understand that the same value of x cannot cannot be matched to two different values of y.

For example, the following set does NOT represent a function, right?

(1, 9), (2, 13), (3, 15), (2, 15), (5, 12), (6, 10)}

In the given set, the number 2 is matched to 13 in the point (2, 13) and to 15 in the point (2, 15). This means the set is not a function.

Am I right here?
 
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  • #2
nycmathguy said:
Homework Statement:: Determine which set is a function.
Relevant Equations:: n/a

Here is the fuzzy definition of a function as presented by Ron Larson.

Definition of Function

A function f from a set A to a set B is a relation that assigns to each element x
in the set A exactly one element y in the set B. The set A is the domain (or set
of inputs) of the function f, and the set B contains the range (or set of outputs).
What makes you think this is a "fuzzy" definition.
nycmathguy said:
Larson goes on to say:

The ordered pairs below can represent a function. The first coordinate (x-value) is
the input and the second coordinate (y-value) is the output.

{(1, 9), (2, 13), (3, 15), (4, 15), (5, 12), (6, 10)}

Let me see.

The above set of elements is a function because every x-value is matched to a unique y-value. Correct?
Yes. A little further on you'll be introduced to the concept of one-to-one functions, for which each y-value is matched by a unique x-value.
The set above represents a function, but not a one-to-one function, because two input values, 3 and 4, are matched to a single output value, 15.
nycmathguy said:
I understand that the same value of x cannot cannot be matched to two different values of y.

For example, the following set does NOT represent a function, right?

(1, 9), (2, 13), (3, 15), (2, 15), (5, 12), (6, 10)}

In the given set, the number 2 is matched to 13 in the point (2, 13) and to 15 in the point (2, 15). This means the set is not a function.

Am I right here?
Yes.
 
  • #3
Mark44 said:
What makes you think this is a "fuzzy" definition.
Yes. A little further on you'll be introduced to the concept of one-to-one functions, for which each y-value is matched by a unique x-value.
The set above represents a function, but not a one-to-one function, because two input values, 3 and 4, are matched to a single output value, 15.
Yes.
I am not too clear on this one-to-one function. I am not there in the textbook.

1. Can you provide an example using a set similar to the one here?

A. What makes a function one-one-one?

B. A function is one-to-one if a passes the vertical line test. Yes? What if a function passes the vertical line test but fails the horizontal line test and vice-versa?

C. What is the basic difference between the vertical line test and the horizontal line test?
 
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  • #4
nycmathguy said:
I am not too clear on this one-to-one function. I am not there in the textbook.

1. Can you provide an example using a set similar to the one here?
Here's almost the same set as in your first example. The only difference is that the ordered pair that was (4, 15) is now (4, 11).
{(1, 9), (2, 13), (3, 15), (4, 11), (5, 12), (6, 10)}
If you plot the 6 points in this set, you will see that the plot passes the vertical line test (it's a function) and the horizontal line test (it's also a one-to-one function).
nycmathguy said:
A. What makes a function one-one-one?
I already explained this:
It's a function "for which each y-value is matched by a unique x-value."
In my example above no y-value is associated with more than one x-value.
nycmathguy said:
B. A function is one-to-one if a passes the vertical line test. Yes? What if a function passes the vertical line test but fails the horizontal line test and vice-versa?
No, you have these backwards. A relation is a function if it passes the vertical line test. This guarantees that each x-value is paired with only one y-value. If a function also passes the horizontal line, it is a one-to-one function.
nycmathguy said:
C. What is the basic difference between the vertical line test and the horizontal line test?
The vertical line test is used to verify that a relation is actually a function. The horizontal line test would be used to verify that a function is also a one-to-one function. Note that if a relation passes the horizontal line test, but doesn't pass the vertical line test, what you have is a bunch of points stacked on top of each other -- not a function.
 

1. What is a function?

A function is a mathematical concept that describes the relationship between two sets of numbers, known as the input and output. It maps each input value to a unique output value, and is often represented as an equation or a graph.

2. How do you determine if a set is a function?

To determine if a set is a function, you must check if each input value has a unique output value. This can be done by creating a mapping table or by graphing the points. If there are no repeated input values with different output values, then the set is a function.

3. What is the difference between a function and a relation?

A function is a type of relation where each input value has a unique output value. A relation, on the other hand, is a set of ordered pairs that can have multiple output values for the same input value. In other words, all functions are relations, but not all relations are functions.

4. Can a set be both a function and a relation?

Yes, a set can be both a function and a relation. If all input values have a unique output value, then the set is a function. If there are any repeated input values with different output values, then the set is a relation.

5. How can you represent a function?

A function can be represented in various ways, such as an equation, a mapping table, a graph, or a verbal description. Each representation provides different information about the function and can be used to solve different types of problems.

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