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Explain whether function f(x) = x2 posses an inverse function or not.

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The function
f (x) = x 2

If we input x =4, then
f (4) = (4) 2= 16.
Similarly, if we input x = -4, then
f (-4) = (-4) 2= 16.
As you can see, two different inputs, 4 and -4, give rise to the same output, 16. This shows that the function f (x) = x 2 is NOT one-to-one, and therefore cannot have an inverse.
Another way to prove to yourself that f (x) = x 2 does not have an inverse is to ask the question, "can we recover -4 from 16?" One way we could recover -4 from 16 is to take - √16. Now suppose we input x = 4 into the same function. f (4) also equals 16. We can recover 4 by taking the square root of 16. This creates a problem; in order to recover the two legitimate input values from the function f (x) = x 2, two different operations were used. In this case, the reverse operation is ambiguous, it could be either √ or -√, showing that the operation of squaring a number is not consistently reversible. Therefore, the function f (x) = x 2 does NOT have an inverse.


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