the output must be a single value

Oct 13, 2025 by windailys.com 8 views

In mathematics, a function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). A valid function must satisfy two main properties: each input must correspond to exactly one output, and the output must be a single value. In this article, we will examine four sets of data, A, B, C, and D, to determine which ones represent valid functions.

What is a Function?

A function is a relation between a set of inputs (domain) and a set of possible outputs (range). It is often represented as a set of ordered pairs, where each pair consists of an input and its corresponding output. For example, the set of data $T=\{(-4,-3),(3,1),(4,5),(8,9),(11,16)\}$ represents a relation between the inputs $-4, 3, 4, 8,$ and $11$ and their corresponding outputs $-3, 1, 5, 9,$ and $16$ .

Property 1: Each Input Must Correspond to Exactly One Output

One of the main properties of a function is that each input must correspond to exactly one output. This means that if we have two ordered pairs, $(a, b)$ and $(a, c)$ , then $b$ must equal $c$ . In other words, if the input $a$ appears twice in the set of data, then the corresponding outputs must be the same.

Property 2: The Output Must be a Single Value

Another important property of a function is that the output must be a single value. This means that if we have an ordered pair $(a, b)$ , then $b$ must be a single value, not a set of values.

Set A: $T=\{(-4,-3),(3,1),(4,5),(8,9),(11,16)\}$

Let's examine the set of data $T=\{(-4,-3),(3,1),(4,5),(8,9),(11,16)\}$ . We can see that each input corresponds to exactly one output, and the output is a single value. Therefore, set A represents a valid function.

Set B: $M=\{(-2,0),(1,2),(5,6),(-2,0),(16,13)\}$

Now, let's examine the set of data $M=\{(-2,0),(1,2),(5,6),(-2,0),(16,13)\}$ . We can see that the input $-2$ appears twice in the set of data, but the corresponding outputs are not the same. Therefore, set B does not represent a valid function.

Set C: $Z=\{(-1,-4),(1,2),(1,6),(8,9),(13,15)\}$

Next, let's examine the set of data $Z=\{(-1,-4),(1,2),(1,6),(8,9),(13,15)\}$ . We can see that the input $1$ appears twice in the set of data, but the corresponding outputs are not the same. Therefore, set C does not represent a valid function.

Set D:

Unfortunately, we do not have enough information to determine whether set D represents a valid function or not.

Conclusion

In conclusion, only set A represents a valid function. Sets B and C do not represent valid functions because they do not satisfy the two main properties of a function: each input must correspond to exactly one output, and the output must be a single value. Set D cannot be determined as a valid function without more information.

References

[1] "Functions" by Khan Academy
[2] "Relations and Functions" by Math Open Reference
[3] "Functions" by Wolfram MathWorld

Q: What is a function?

A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). It is often represented as a set of ordered pairs, where each pair consists of an input and its corresponding output.

Q: What are the two main properties of a function?

The two main properties of a function are:

Each input must correspond to exactly one output: If we have two ordered pairs, $(a, b)$ and $(a, c)$ , then $b$ must equal $c$ . In other words, if the input $a$ appears twice in the set of data, then the corresponding outputs must be the same.
The output must be a single value: If we have an ordered pair $(a, b)$ , then $b$ must be a single value, not a set of values.

Q: How do I determine if a set of data represents a valid function?

To determine if a set of data represents a valid function, you need to check if it satisfies the two main properties of a function:

Each input must correspond to exactly one output: Check if each input appears only once in the set of data.
The output must be a single value: Check if each output is a single value, not a set of values.

Q: What is the difference between a function and a relation?

A function is a relation between a set of inputs (domain) and a set of possible outputs (range) that satisfies the two main properties of a function. A relation is a set of ordered pairs, but it does not necessarily satisfy the two main properties of a function.

Q: Can a function have multiple outputs for the same input?

No, a function cannot have multiple outputs for the same input. If a function has multiple outputs for the same input, then it is not a valid function.

Q: Can a function have no outputs for a given input?

Yes, a function can have no outputs for a given input. In this case, the input is said to be undefined or not in the domain.

Q: Can a function have multiple inputs with the same output?

Yes, a function can have multiple inputs with the same output. In this case, the function is said to be many-to-one.

Q: Can a function have no inputs?

Yes, a function can have no inputs. In this case, the function is said to be empty.

Q: Can a function have multiple outputs for different inputs?

Yes, a function can have multiple outputs for different inputs. In this case, the function is said to be one-to-many.

Q: Can a function be represented as a table?

Yes, a function can be represented as a table. In this case, the table is called a function table.

Q: Can a function be represented as a graph?

Yes, a function can be represented as a graph. In this case, the graph is called a function graph.

Q: Can a function be represented as an equation?

Yes, a function can be represented as an equation. In this case, the equation is called a function equation.

Conclusion

In conclusion, functions are an important concept in mathematics that can be represented in various ways, including as a set of ordered pairs, a table, a graph, or an equation. By understanding the properties of a function, you can determine if a set of data represents a valid function or not.

References

[1] "Functions" by Khan Academy
[2] "Relations and Functions" by Math Open Reference
[3] "Functions" by Wolfram MathWorld