Let’s solve the problem step by step.
We are given a relation:
a = {(-7, -2), (-3, 5), (5, 1), (2, 1)}
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Step 1: Find the inverse of the relation
To find the inverse of a relation, we
swap the x and y values in each ordered pair.
Original pairs:
- (-7, -2) → becomes (-2, -7)
- (-3, 5) → becomes (5, -3)
- (5, 1) → becomes (1, 5)
- (2, 1) → becomes (1, 2)
So the inverse is:
{(-2, -7), (5, -3), (1, 5), (1, 2)}
✔ That matches what’s written on the paper — good!
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Step 2: Determine if it’s a one-to-one function
A function is
one-to-one if
each input has exactly one output, AND
each output comes from exactly one input. In other words, no two different inputs can have the same output.
But here’s the key: We’re being asked whether the
original relation is a one-to-one function.
Wait — first, is the original relation even a function?
Check: Does any x-value repeat with a different y-value?
Original relation:
x-values: -7, -3, 5, 2 → all unique → so yes, it
is a function.
Now, is it
one-to-one? Check if any y-values repeat.
y-values: -2, 5, 1, 1 → Oh! The y-value
1 appears twice: for x=5 and x=2.
That means two different inputs (5 and 2) give the same output (1). So it’s
not one-to-one.
Also, look at the inverse: {(−2, −7), (5, −3), (1, 5), (1, 2)}
In the inverse, the input
1 maps to both
5 and
2 → which means the inverse is
not a function. And if the inverse isn’t a function, then the original function is not one-to-one.
✔ So the answer “Yes” written on the paper is
incorrect.
The correct answer should be:
No, it is not a one-to-one function.
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Final Answer:
No
Parent Tip: Review the logic above to help your child master the concept of inverse functions and relations worksheet.
