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Range of a function

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Find the range of \[\sqrt{f(x)-g(x)} - \sqrt{f(x)+g(x)}\]

Such that range of \(f(x)\) is \((a,b)\) for all \(x \in R\)

And range of \(g(x)\) is \((c,d)\) \(x \in R\)

Where \[ a,c \in R^{-} \space and \space b,d \in R^{+}\]

\(f(x)\) and \(g(x)\) are continous and both are increasing functions.

Note by Kushal Patankar
1 year, 8 months ago

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Don't you think your question is too vague?? Ninad Akolekar · 1 year, 8 months ago

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@Ninad Akolekar Yep, i got that now. It was just crazy curiosity Kushal Patankar · 1 year, 8 months ago

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This question is too vague. We cannot proceed further without some information about the functions given . please try to expkain further. Raghav Vaidyanathan · 1 year, 8 months ago

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@Raghav Vaidyanathan What would be further necessary information. Kushal Patankar · 1 year, 8 months ago

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@Kushal Patankar We do not know anything about the fuctions. First of all we dont know whether they are continuous. Secondly, we dont know what values they take. For example assume f() takes some value in its range for some x, we dont know the respective value of g() for that x.. It may even be possible for the given expression to become undefined. But we cant say anything since we dont know f and g. If f and g are given, then it is possible.. Otherwise, i dont think we can do it.

We may be able to devise a set method or tactics to solve problems like this when f and g are given. We may be able to find a weak range.. Nothing more. Raghav Vaidyanathan · 1 year, 8 months ago

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@Raghav Vaidyanathan I added some information and hope thats enough Kushal Patankar · 1 year, 8 months ago

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@Kushal Patankar Nope.. Even now, it is ambiguous. Consider something like f(x)=x and g(x)=x+1 for x between some negative number and some positive number. Here, there is no element in range as given expression is undefined everywhere. Raghav Vaidyanathan · 1 year, 8 months ago

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Does it matter whether \(f(x)\) and \(g(x)\) are continous? Kushal Patankar · 1 year, 8 months ago

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