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how to show a function is invertible

say f(x)=(4x^3)/((x^2) + 1)

how can i show f has an inverse?

i understand that for a function to be invertible, f(x1) does not equal f(x2) whenever x1 does not equal x2. but im unsure how i can apply it to the above function. help please, thanks :smile:
Reply 1
Avatar for SimonM
SimonM
18
Well, there are many ways to prove that a function is injective and hence has the inverse you seek. One of them is to show that the function is increasing (Can you see why)
Reply 2
Avatar for Booyah!
Booyah!
OP
no not really...
Reply 3
Avatar for Kolya
Kolya
17
If the function is strictly increasing then f(x2)>f(x1)f(x_2) > f(x_1) whenever x2>x1x_2 > x_1. You should be able to see that this implies the function is also injective.
Reply 4
Avatar for Booyah!
Booyah!
OP
Kolya
If the function is strictly increasing then f(x2)>f(x1)f(x_2) > f(x_1) whenever x2>x1x_2 > x_1. So, clearly, f(x1)f(x_1) cannot equal f(x2)f(x_2) whenever x1≠x2x_1 \neq x_2. Hence strictly increasing means the function is injective.


oh i get it now, but is that enough proof? would i need to give some values or something?
Reply 5
Avatar for Booyah!
Booyah!
OP
ignore that, thanks :smile:
Reply 6
Avatar for Kolya
Kolya
17
Within the context of the question, I think it would be enough to just state that strictly increasing implies injective, and then use that idea to complete the question. (However, the proof of the intermediate result is easy.)

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