For what value(s) of k will the relation not be a function?
A = {(1.5k−4, 7), (−0.5k+8, 15)}
The Correct Answer and Explanation is:
To determine the value(s) of ( k ) for which the given relation ( A = {(1.5k – 4, 7), (-0.5k + 8, 15)} ) is not a function, we must first recall the definition of a function. A relation is a function if each input (or domain element) corresponds to exactly one output (or range element).
In this case, the relation ( A ) consists of two ordered pairs: ( (1.5k – 4, 7) ) and ( (-0.5k + 8, 15) ). For this to be a function, the first elements (inputs) of the pairs must be distinct. If they are not distinct, the relation will fail to be a function, as it would violate the rule that each input should correspond to only one output.
Step 1: Set the inputs equal to each other
The inputs of the two ordered pairs are ( 1.5k – 4 ) and ( -0.5k + 8 ). For the relation to not be a function, these two inputs must be equal, as that would imply the same input leads to different outputs, which breaks the definition of a function.
Set the two inputs equal to each other:
[
1.5k – 4 = -0.5k + 8
]
Step 2: Solve for ( k )
First, get all the terms involving ( k ) on one side:
[
1.5k + 0.5k = 8 + 4
]
[
2k = 12
]
Now, solve for ( k ):
[
k = \frac{12}{2} = 6
]
Step 3: Conclusion
For ( k = 6 ), the two inputs become equal, meaning the relation is not a function because the input ( 1.5(6) – 4 = 9 – 4 = 5 ) is the same for both ordered pairs, while the corresponding outputs are different (7 and 15). Therefore, the value of ( k ) that makes the relation not a function is ( k = 6 ).
Thus, the answer is ( k = 6 ).