Which Relation Is A Function Of $x$?$\[ \begin{array}{|c|c|} \hline x & Y \\ \hline -1 & 7 \\ \hline 2 & -9 \\ \hline 2 & 8 \\ \hline 3 & -4 \\ \hline \end{array} \\]$\[ \begin{array}{|c|c|} \hline x & Y \\ \hline -8 & -9

Mar 1, 2025 by ADMIN 222 views

Introduction

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 function is a special type of relation where each input is associated with exactly one output. In this article, we will explore which relation is a function of $x$ based on the given data.

Understanding Functions

A function is a relation that satisfies the following conditions:

Each input is associated with exactly one output.
Each output is associated with exactly one input.

In other words, a function is a relation where each value of $x$ is mapped to a unique value of $y$ .

Analyzing the Given Relations

We are given two relations in the form of tables:

$x$	$y$
-1	7
2	-9
2	8
3	-4
$x$	$y$
---	---
-8	-9

Relation 1

Let's analyze the first relation:

$x$	$y$
-1	7
2	-9
2	8
3	-4

In this relation, we can see that the value of $x$ is repeated for two different values of $y$ . Specifically, the value of $x$ is 2, and the corresponding values of $y$ are -9 and 8. This means that the relation does not satisfy the condition of a function, where each input is associated with exactly one output.

Relation 2

Now, let's analyze the second relation:

$x$	$y$
-8	-9

In this relation, we can see that the value of $x$ is -8, and the corresponding value of $y$ is -9. There are no repeated values of $x$ , and each value of $x$ is associated with a unique value of $y$ . This means that the relation satisfies the condition of a function.

Conclusion

Based on the analysis of the given relations, we can conclude that the second relation is a function of $x$ . The first relation does not satisfy the condition of a function, as the value of $x$ is repeated for two different values of $y$ .

Key Takeaways

A function is a relation where each input is associated with exactly one output.
A function is a special type of relation that satisfies the condition of each input being associated with exactly one output.
The second relation is a function of $x$ , while the first relation is not.

Q: What is a function in mathematics?

A: A function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. A function is a special type of relation where each input is associated with exactly one output.

Q: What are the conditions for a relation to be a function?

A: A relation is a function if and only if each input is associated with exactly one output. In other words, a function is a relation where each value of $x$ is mapped to a unique value of $y$ .

Q: How do I determine if a relation is a function?

A: To determine if a relation is a function, you need to check if each input is associated with exactly one output. You can do this by looking for repeated values of $x$ and checking if they are associated with the same or different values of $y$ .

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

A: A function is a special type of relation where each input is associated with exactly one output. A relation, on the other hand, is a set of ordered pairs that may or may not satisfy the condition of a function.

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

A: Yes, a relation can have multiple outputs for the same input. However, if a relation has multiple outputs for the same input, it is not a function.

Q: Can a function have multiple inputs that map to the same output?

A: Yes, a function can have multiple inputs that map to the same output. For example, the function $f(x) = 2$ has multiple inputs (all values of $x$ ) that map to the same output (2).

Q: How do I graph a function?

A: To graph a function, you need to plot the points on a coordinate plane that satisfy the function. You can use a table of values or a graphing calculator to help you plot the points.

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for the function. In other words, it is the set of all values of $x$ that the function can accept.

Q: What is the range of a function?

A: The range of a function is the set of all possible output values for the function. In other words, it is the set of all values of $y$ that the function can produce.

Q: Can a function have an empty domain or range?

A: Yes, a function can have an empty domain or range. For example, the function $f(x) = 1/x$ has an empty domain when $x = 0$ , and the function $f(x) = x^2$ has an empty range when $y < 0$ .

Q: How do I find the inverse of a function?

A: To find the inverse of a function, you need to swap the $x$ and $y$ values and solve for $y$ . This will give you the inverse function.

Q: What is the difference between a one-to-one function and a many-to-one function?

A: A one-to-one function is a function where each input is associated with exactly one output. A many-to-one function is a function where multiple inputs are associated with the same output.

Q: Can a function be both one-to-one and many-to-one?

A: No, a function cannot be both one-to-one and many-to-one. A function is either one-to-one or many-to-one, but not both.

Q: How do I determine if a function is one-to-one or many-to-one?

A: To determine if a function is one-to-one or many-to-one, you need to check if each input is associated with exactly one output. If each input is associated with exactly one output, the function is one-to-one. If multiple inputs are associated with the same output, the function is many-to-one.