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Question
Which of the following statements is true about the above four relations?
Which of the following statements is true about the above four relations?
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Solution
The correct option is B R does not represent a function and the relation rule is y=x.
Detailed step-by-step solution:
We know:
1. If all the inputs are not engaged in a relation from x to y, then the relation does not qualify as a function.
2. Every input should be related to exactly one output. If that’s not the case, then a relation will not qualify as a function.
Let’s consider all the relations one-by-one along with the options.
Relation P: The input and output pairs are: (−2,5),(−1,2),(0,1),(3,10)
Here for each input, there is a unique output.
As it is following the conditions of a function, the relation P is a function.
According to option A, the given functional rule is y=−2x+1.
Let’s check with the inputs.
For x=−2,y=−2x+1=−2×(−2)+1 (substituting for x=−2)
=4+1=5, which is the same as the given output.
For x=−1,y=−2x+1=−2×(−1)+1 (substituting for x=−1)
=2+1=3
Here, the output is 3, but the output given in the relation is 2.
So it is not following the given functional rule.
So, the statement in option A is not true.
Relation Q: The input and output pairs are: (2,−2),(2,2),(1,−1),(1,1),(3,−3)
Here for some inputs, there are more than one output.
As it is not following the conditions of a function, the relation Q is not a function.
According to option B, the given relation rule is y=−x.
Let’s check with the inputs.
For x=2,y=−x=−2.
For x=1,y=−x=−1.
For x=0,y=−x=−0=0.
Here, for the input of 2, the output is obtained as −2, but the output given in the relation is 2 and −2.
So it is not following the given relation rule.
So, the statement in option B is not true.
Relation R: The input and output pairs are: (−2,−2),(−1,−1),(0,0),(1,1)
Here for each input, there is a unique output.
But all the inputs do not have relation, as there is no output for 2.
As it is not following the conditions of a function, so the relation R is not a function.
According to option C, the given relation rule is y=x.
Let’s check with the inputs.
For x=−2,y=x=−2.
For x=−1,y=x=−1.
For x=0,y=x=0.
For x=1,y=x=1.
Here, it is following the given relation rule.
So, the statement in option C is true.
Relation S: The input and output pairs are: (−2,−2),(−1,−1),(0,0),(1,1),(2,−2)
Here for each input, there is a unique output.
As it is following the conditions of a function, so the relation S is a function.
According to option D, the given functional rule is y=x.
Let us check with the inputs.
For x=−2,y=x=−2, which is the same as the given output.
For x=−1,y=x=−1, which is the same as the given output.
For x=0,y=x=0, which is the same as the given output.
For x=1,y=x=1, which is the same as the given output.
For x=2,y=x=2.
Here, the output is coming as 2, but the output given in the relation is −2.
So it is not following the given functional rule.
So, the statement in option D is not true.
So, option C is the correct answer.
Detailed step-by-step solution:
We know:
1. If all the inputs are not engaged in a relation from x to y, then the relation does not qualify as a function.
2. Every input should be related to exactly one output. If that’s not the case, then a relation will not qualify as a function.
Let’s consider all the relations one-by-one along with the options.
Relation P: The input and output pairs are: (−2,5),(−1,2),(0,1),(3,10)
Here for each input, there is a unique output.
As it is following the conditions of a function, the relation P is a function.
According to option A, the given functional rule is y=−2x+1.
Let’s check with the inputs.
For x=−2,y=−2x+1=−2×(−2)+1 (substituting for x=−2)
=4+1=5, which is the same as the given output.
For x=−1,y=−2x+1=−2×(−1)+1 (substituting for x=−1)
=2+1=3
Here, the output is 3, but the output given in the relation is 2.
So it is not following the given functional rule.
So, the statement in option A is not true.
Relation Q: The input and output pairs are: (2,−2),(2,2),(1,−1),(1,1),(3,−3)
Here for some inputs, there are more than one output.
As it is not following the conditions of a function, the relation Q is not a function.
According to option B, the given relation rule is y=−x.
Let’s check with the inputs.
For x=2,y=−x=−2.
For x=1,y=−x=−1.
For x=0,y=−x=−0=0.
Here, for the input of 2, the output is obtained as −2, but the output given in the relation is 2 and −2.
So it is not following the given relation rule.
So, the statement in option B is not true.
Relation R: The input and output pairs are: (−2,−2),(−1,−1),(0,0),(1,1)
Here for each input, there is a unique output.
But all the inputs do not have relation, as there is no output for 2.
As it is not following the conditions of a function, so the relation R is not a function.
According to option C, the given relation rule is y=x.
Let’s check with the inputs.
For x=−2,y=x=−2.
For x=−1,y=x=−1.
For x=0,y=x=0.
For x=1,y=x=1.
Here, it is following the given relation rule.
So, the statement in option C is true.
Relation S: The input and output pairs are: (−2,−2),(−1,−1),(0,0),(1,1),(2,−2)
Here for each input, there is a unique output.
As it is following the conditions of a function, so the relation S is a function.
According to option D, the given functional rule is y=x.
Let us check with the inputs.
For x=−2,y=x=−2, which is the same as the given output.
For x=−1,y=x=−1, which is the same as the given output.
For x=0,y=x=0, which is the same as the given output.
For x=1,y=x=1, which is the same as the given output.
For x=2,y=x=2.
Here, the output is coming as 2, but the output given in the relation is −2.
So it is not following the given functional rule.
So, the statement in option D is not true.
So, option C is the correct answer.
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