You need to show that the function f: \mathbb{R} \rightarrow \mathbb{R} defined by f(x) = x^2 is not one-to-one (injective) but is many-to-one.
Answer (2)
The function f ( x ) = x 2 is not one-to-one because different inputs can produce the same output. For example, both 2 and − 2 yield 4 . This shows that the function maps multiple inputs to a single output, confirming it is many-to-one.
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To show that the function f : R → R defined by f ( x ) = x 2 is not one-to-one (injective) but is many-to-one, we'll need to explore what these terms mean and how they apply to this function.
One-to-one Function:
A function is one-to-one (injective) if different inputs lead to different outputs, or in formal terms, for all a , b ∈ R , if f ( a ) = f ( b ) , then it must be the case that a = b .
Testing f ( x ) = x 2 for Injectiveness:
Consider two different real numbers, a and b , such that a = b . The function f ( x ) = x 2 gives:
f ( a ) = a 2 and f ( b ) = b 2
If f ( a ) = f ( b ) , then a 2 = b 2 . This means that a = b or a = − b . Therefore, f ( a ) = f ( b ) does not necessarily imply a = b . For example, f ( 2 ) = 4 and f ( − 2 ) = 4 , but 2 = − 2 . Thus, the function is not one-to-one.
Many-to-one Function:
A many-to-one function means that a single output value is the result of more than one distinct input value. Since we demonstrated above that both x = a and x = − a can yield the same output a 2 , f ( x ) = x 2 is indeed a many-to-one function.
Conclusion:
Since f ( x ) = x 2 is not injective (one-to-one) because it maps both x = a and x = − a to the same output a 2 , it is a many-to-one function. This illustrates that for certain functions, multiple inputs can indeed produce the same output, classifying them as many-to-one.
