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Persisting doubts

I am still confused with the problem Simply Exponential \[\quad \quad \quad \quad { x }^{ x }=x\\ \Longrightarrow x={ x }^{ 1/x }\\ \Longrightarrow { x }^{ x }={ x }^{ 1/x }\\ \Longrightarrow x\ln { x } =\frac { 1 }{ x } \ln { x } \\ \Longrightarrow \ln { x } (\frac { { x }^{ 2 }-1 }{ x } )=0\ \\ \Longrightarrow x=\pm 1(Correct \quad answer)\] Now,since we have included a step \(x\ln { x } =\frac { 1 }{ x } \ln { x }\) in our solution,according to the domain of the logarithmic function, x should be positive,thus omitting -1.Then why -1(as we cannot take log of a negative number)?

Then, I am confused that the given expression(actually an equation) is a function or not. I mean, if it is given that \(f\left( x \right) ={ x }^{ x }-x\) and then we are asked to find the roots or zeroes of the function, then the answer will be only +1 or still be both+1 & -1?(I think that it should be only +1 as the domain of \(x^{x}\) is \(x>0\) )

Also, I don't know why, but is the second step correct and safe?

Please clear my doubts,and please provide any kind of guidance wherever you think is needed. Thanks

Note by Abhijeet Verma
1 year, 6 months ago

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You are right that, if we just want to work with continuous functions over the field of real numbers, the domain of both \(x^{x}\) and \(\ln(x)\) is \(x \gt 0.\) So the step where we take the natural log of both sides can only be done under the condition that \(x \gt 0,\) and hence with this process only the solution \(x = 1\) is valid.

However, the function \(f(x) = x^{x}\) is also purely real, (i.e., has an imaginary component of \(0*i\)), whenever \(x\) is a negative integer, and since \((-1)^{-1} = \dfrac{1}{(-1)^{1}} = -1\) we also have that \(x = -1\) is a solution to the equation \(x^{x} = x.\)

So while \(g(x) = x^{x} - x\) is a continuous function only for \(x \gt 0,\) it does still have a "zero" at \(x = -1.\) Brian Charlesworth · 1 year, 6 months ago

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@Brian Charlesworth Sir, if -1 is a zero of the function,then what will be it's(I mean the function's) domain? (As the zero sholud lie in it's domain).

Will it's domain be (all positive real numbers + negative integers)? Also, according to you, -1 is not a solution which we get mathematically(I mean analytically) and it is just by random hit and trial, so if in a subjective paper, we have to conclude -1(it is a bit funny and pointless, but just...),what's the best way to do it? Abhijeet Verma · 1 year, 6 months ago

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@Abhijeet Verma I was wondering that myself. WolframAlpha lists the domain only as \(x \gt 0,\) and yet it also states that \(x = -1\) is a root, which at first glance seems like a contradiction; how can a function have a root outside of its domain? The only way I can resolve this is that as a continuous function its domain is \(x \gt 0,\) and that \(x = 1\) is technically the only root, but it also happens to be the case that \(g(-1) = 0.\) I'm not sure if we should really call \(x = -1\) a root, (or zero), but nevertheless \(x = -1\) is a valid solution to the equation \(x^{x} = x.\) So for now I would still consider the domain as just \(x \gt 0,\) and that the solution \(x = -1\) is, as you say, a hit and trial "anomaly".

When dealing with powers of negative numbers it makes more sense to work in the field of complex numbers. That way we can expand the domain and such anomalies then become "analytic" solutions. Brian Charlesworth · 1 year, 6 months ago

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@Brian Charlesworth Wolfram doesn't deal well with isolated point domains (even if you think that they are nice like the integers). Calvin Lin Staff · 1 year, 6 months ago

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@Calvin Lin So can we then legitimately choose that the domain of \(g(x) = x^{x} - x\) is the set of all positive reals as well as all negative integers? Then both \(1\) and \(-1\) would be valid roots. If we wanted to deal only with a continuous function then we could restrict the domain to "just" \(x \gt 0.\) Brian Charlesworth · 1 year, 6 months ago

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@Brian Charlesworth Correct.

Of course, for a function to be completely well defined, the domain needs to be stated. In the event that the domain isn't stated, it is extremely open to interpretation. E.g. What are the zeros of \( f(x) = x^2 + 1 \)? Those who say no zeros (working in the domain of real numbers) are equally correct as those who say zero of \(i \) and \( -i \) (working in the domain of complex numbers).

In the event that the domain is over specified, then we simply ignore values where the function doesn't make sense. In this case, it means that we remove those negative non-integer values. Calvin Lin Staff · 1 year, 6 months ago

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@Calvin Lin O.k., great. Thanks for clearing that up. :) Brian Charlesworth · 1 year, 6 months ago

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See Ivan's solution, which avoids the numerous issues that you have raised.

Whenever we manipulate an equation, we have to be careful about the steps that we're taking, and check to see if we're introducing extraneous solutions, or losing actual solutions. For example, by taking logs, we need to assume that \( x > 0 \). Calvin Lin Staff · 1 year, 6 months ago

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Akhil Bansal · 1 year, 6 months ago

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@Janardhanan Sivaramakrishnan Sir & @Brian Charlesworth Sir please help. Abhijeet Verma · 1 year, 6 months ago

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@Abhijeet Verma \(x \ln x = \frac{1}{x} \ln x\)

Let \(x=r e^{i\theta}\). Then, \(\ln x = \ln r + i\theta\) (This step makes it possible to have natural logarithms for negative or even complex numbers).

\((\ln r + i\theta)(r e^{i\theta} - r^{-1} e^{-i\theta})=0\)

Case 1 : \(\ln r = 0\) and \(\theta = 0\). This implies \(x=1\)

Case 2 : \(r e^{i\theta} = r^{-1} e^{-i\theta}\)

This means \(r^2 e^{2i\theta} = 1\)

Which in turn means that \(|r| =1\) and \(2\theta = 2n\pi\), or \(\theta = 0,\pi\)

This gives \(x=1\) again. But, we also find the second solution \(x=-1\). Janardhanan Sivaramakrishnan · 1 year, 6 months ago

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@Janardhanan Sivaramakrishnan Thanks Sir for clearing the confusion. Abhijeet Verma · 1 year, 6 months ago

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I think \( x = +1\) is the only correct solution because domain of our function is \( x > 0 \) Akhil Bansal · 1 year, 6 months ago

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@Akhil Bansal It is not stated that the domain is \( x > 0 \). It asks for "distinct real solutions". Calvin Lin Staff · 1 year, 6 months ago

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@Calvin Lin If some function has domain \(x > 0\) ,then how could it have a negative solution?
Before finding the solution of some function, don't we first check its domain? Akhil Bansal · 1 year, 6 months ago

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@Akhil Bansal I think that we're dealing with two distinct questions. If we are just looking for solutions to the equation \(x^{x} = x\) then both \(1\) and \(-1\) are valid solutions, and thus the answer to the question Abhijeet has linked to is \(2.\)

If, however, we were asked for the number of roots of the function \(f(x) = x^{x} - x,\) then we would have be mindful of the domain. If we establish the domain as \(x \gt 0\) then there would be only one valid root, namely \(x = 1.\) If we were to also include the set of all negative numbers in the domain then \(x = -1\) would also be a valid root. The more "standard" choice of domain would be \(x \gt 0,\) though, since \(f(x)\) would then be continuous and differentiable over all the domain. Brian Charlesworth · 1 year, 6 months ago

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@Akhil Bansal Thanks for inserting the graph. Yes, analytically,only +1 is the solution or zero of the function(still, -1 is confusing as \(f(-1)=0\)),as mentioned by Brian Sir, but -1 is a valid solution of the equation \(x^{x}=x\) [if not considering it a function],which we get by hit & trial. Abhijeet Verma · 1 year, 6 months ago

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@Abhijeet Verma From where do you get this question? Akhil Bansal · 1 year, 6 months ago

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@Akhil Bansal I have mentioned the link in the note itself. Abhijeet Verma · 1 year, 6 months ago

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@Abhijeet Verma There can be a big debate on this self-contradicting question. Akhil Bansal · 1 year, 6 months ago

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@Akhil Bansal What surprised me was that the all the reports were marked as resolved. Abhijeet Verma · 1 year, 6 months ago

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@Abhijeet Verma The reports were resolved because they are not valid. The main reasons were:

  • They did not justify that there is only 1 solution.
  • They did not read the question and see that it asks for the total number of solutions (esp those that say \( 2^ 2 \neq 2 \)).
  • Claiming that -1 is not a solution, when it does satisfy the conditions.
  • Their steps were invalid.
Calvin Lin Staff · 1 year, 6 months ago

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@Calvin Lin Surely sir, I meant that seeing all reports as resolved made me think again and then i got a bit confused.

BTW, Sir, can you please go through my note and comments and tell if -1 will be a zero of the function or not. Thanks Abhijeet Verma · 1 year, 6 months ago

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@Abhijeet Verma It is a zero of the function since \( f(-1)= 0 \). Calvin Lin Staff · 1 year, 6 months ago

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@Calvin Lin Thanks sir for clearing the confusion. Abhijeet Verma · 1 year, 6 months ago

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