For what value of $c$ is the relation a function?
$\left\{(2,8),(12,3),(c, 4),(-1,8),(0,3)\right\}$
Answer (2)
A relation is a function if each x-value maps to a unique y-value.
Check the given values for c to see if any result in the same x-value mapping to different y-values.
If c = − 1 , the relation contains ( − 1 , 4 ) and ( − 1 , 8 ) , which is not a function.
The only value of c that makes the relation a function is 1 .
Explanation
Understanding the Problem We are given a relation as a set of ordered pairs:
{( 2 , 8 ) , ( 12 , 3 ) , ( c , 4 ) , ( − 1 , 8 ) , ( 0 , 3 )} .
We need to find the value of c for which this relation is a function. A relation is a function if each element in the domain (the first element of each ordered pair) maps to a unique element in the range (the second element of each ordered pair). In simpler terms, no two ordered pairs can have the same first element but different second elements.
Identifying Existing x-values Let's examine the given ordered pairs and identify the existing x -values: 2 , 12 , c , − 1 , and 0 . For the relation to be a function, c cannot be equal to any of the existing x -values: 2 , 12 , − 1 , or 0 .
Checking Possible Values for c Now, let's check the given possible values for c : − 1 , 1 , 2 , 12 .
If c = − 1 , the relation would contain ( − 1 , 4 ) and ( − 1 , 8 ) . Since the x -value − 1 maps to two different y -values ( 4 and 8 ), this violates the function property.
If c = 1 , the relation becomes {( 2 , 8 ) , ( 12 , 3 ) , ( 1 , 4 ) , ( − 1 , 8 ) , ( 0 , 3 )} . Since 1 is not equal to 2 , 12 , − 1 , or 0 , the relation is a function.
If c = 2 , the relation would contain ( 2 , 4 ) and ( 2 , 8 ) . Since the x -value 2 maps to two different y -values ( 4 and 8 ), this violates the function property.
If c = 12 , the relation would contain ( 12 , 4 ) and ( 12 , 3 ) . Since the x -value 12 maps to two different y -values ( 4 and 3 ), this violates the function property.
Conclusion Therefore, the only value of c that makes the relation a function is 1 .
Examples
Imagine you are assigning tasks to employees. Each employee (x-value) can only be assigned one specific task (y-value) at a time to ensure smooth workflow. If an employee is assigned two different tasks simultaneously, it would create confusion and inefficiency. Similarly, in mathematics, for a relation to be a function, each input (x-value) must have a unique output (y-value). This concept is crucial in various real-life applications, such as computer programming, data analysis, and engineering, where ensuring a unique output for each input is essential for accurate and reliable results.
The relation is a function when the value of c is 1 , as it does not conflict with any existing x-values. For values such as − 1 , 2 , or 12 , the relation would not be a function because it would cause repeated x-values with different y-values. Therefore, the answer is c = 1 .
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