Choose the option that best describes how to determine if a relation is a function.
A. Check that for each value of x, there is only one value of y.
B. Check that for each value of y, there is only one value of x.
C. Check that for each output, there is only one input.
D. Check that a horizontal line could move across the y-axis and only intersect the graph of the relation in one spot at a time
Answer (2)
To determine if a relation is a function:
Check if each x-value has only one corresponding y-value.
Option 1 correctly describes this fundamental property of a function.
Options 2 and 3 describe a one-to-one function, a special type of function.
Option 4 describes the horizontal line test, used to determine if the inverse of a function is also a function.
The correct answer is: Check that for each value of x, there is only one value of y.
Explanation
Analyzing the Options To determine if a relation is a function, we need to check the relationship between the input values (x) and the output values (y). A relation is a function if each input (x-value) is associated with only one output (y-value). Let's analyze the given options:
Option 1: Check that for each value of x, there is only one value of y. This statement accurately describes the definition of a function.
Option 2: Check that for each value of y, there is only one value of x. This describes a one-to-one function, which is a special type of function, but not the general definition of a function.
Option 3: Check that for each output, there is only one input. This is equivalent to option 2 and describes a one-to-one function.
Option 4: Check that a horizontal line could move across the y-axis and only intersect the graph of the relation in one spot at a time. This is the horizontal line test, which is used to determine if the inverse of a function is also a function (i.e., if the function is one-to-one).
Identifying the Correct Definition The correct option is the one that accurately defines a function. Option 1 states that for each value of x, there is only one value of y. This is the fundamental definition of a function.
Conclusion Therefore, the best description of how to determine if a relation is a function is to check that for each value of x, there is only one value of y.
Examples
In real life, functions are used to model relationships between different quantities. For example, the relationship between the number of hours worked and the amount of money earned can be represented by a function. If each hour worked corresponds to a unique amount of money earned, then the relationship is a function. Understanding functions helps us predict and analyze these relationships in various fields like economics, physics, and computer science.
To determine if a relation is a function, check that for each value of x, there is only one corresponding value of y. This is the fundamental definition of a function. Hence, the best option is A.
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