math pre calc help?

Question

determine whether the function has an inverse function and if it doesnt then why not? F(x)= {x+3, x<0
{6-x, x> or equal to 0

my book says it doesnt but i cant figure out why

Answer 1

Well in order for it to have an inverse, it would have to be the case that no matter what output you take, you can find an input for it. This is because every function must take EVERY number in its domain and map it to something. If you can find an output of f that isn't mapped to by anything, then if you try to take the inverse function that goes from the target, you'll find some inputs are missing and the definition of a function isn't satisfied.

So consider the number f(x) = 7. You would have to find a number x less than 0 such that x + 3 = 7, or a number greater than or equal to 0 such that 6 - x = 7. We can easily see that for the first case, we'd need x = 4, which is not negative, and for the second case we'd need x = -1, which is not greater than or equal to 0.

Thus we've found our counterexample, so there's no inverse function.

Answer 2

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