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A Hard Function Problem

Based on 2013 Olympiad Problem

A function \(f\) is defined for all \(x\) and has the following property,

\(\large f(x) = \frac{ax+b}{cx+d}\)

If \(a, b, c\) and \(d\) are real number and the function above satisfy \(f(19)=19, f(97) = 97, f(f(x)) = x,\) for every \(x\) value, except \(\large -\frac{d}{c}\), find range of the function

Note by Jason Chrysoprase
3 months, 3 weeks ago

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I'll assume that the domain is \(\mathbb{R} - \{-\frac{d}{c} \} \)

If \(\dfrac{b}{a} \neq \dfrac{d}{c} \), then the range is \(\mathbb{R}-\{\frac{a}{c}\}\) if \(c \neq 0\) and \(\mathbb{R}\) otherwise.

If equality holds above, then \(f(x)\) is a constant function, which contradicts the fact that \(f(19) \neq f(97)\).

In conclusion, the range is \(\mathbb{R}-\{\frac{a}{c}\}\) if \(c \neq 0\) otherwise \(\mathbb{R}\).


Edit:

After finding \(a,b, c, d\),

Either \(f(x)=x\) or \(f(x)=\dfrac{58x-19 \times 97}{x-58} \)

So, the range is either \(\mathbb{R}\) or \(\mathbb{R} - \{58\}\) Deeparaj Bhat · 3 months, 3 weeks ago

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@Deeparaj Bhat Domain

\({x\in \mathbb{R} : x \neq 58}\)

Range

\({f \in \mathbb{R} : f \neq 58}\) Jason Chrysoprase · 3 months, 2 weeks ago

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@Deeparaj Bhat The answer is \(\mathbb{R} - {58}\) since there is no \(x\) value for \(58\) Jason Chrysoprase · 3 months, 2 weeks ago

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@Deeparaj Bhat I don't know functions well , can you please explain a bit more on how you took the range? Rishabh Tiwari · 3 months, 3 weeks ago

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@Deeparaj Bhat So every \(x\) we put, the answer is always real ?

So how about \(f(0)\)? Jason Chrysoprase · 3 months, 3 weeks ago

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@Jason Chrysoprase Oops, sorry for late reply. Whats the problem if we put \(f(0)\).
\(f(0) = \dfrac{a×0 + b}{c×0 + d} = \dfrac{b}{d}\).
Since \(\dfrac{-d}{c}\) is removed from its domain, the answer \(\dfrac{b}{d}\) is real too. Ashish Siva · 3 months, 3 weeks ago

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Check this out Calvin Lin Staff · 3 months, 3 weeks ago

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@Calvin Lin Wow, that guy really wrote the same problem. We both came from Indonesia :) Jason Chrysoprase · 3 months, 3 weeks ago

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@Calvin Lin Sir, Can you please suggest me where can I learn about functions & solving these type of questions? Honestly, I don't know much about this topic , please help me, thank you.! Rishabh Tiwari · 3 months, 3 weeks ago

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@Rishabh Tiwari Practice the challenge quizzes in the chapter and read the wikis. Calvin Lin Staff · 3 months, 2 weeks ago

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@Calvin Lin Thank you sir :-) Rishabh Tiwari · 3 months, 2 weeks ago

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@Jason Chrysoprase why not me??? Ayush Rai · 3 months, 3 weeks ago

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@Ayush Rai Sorry, forgot u :( Jason Chrysoprase · 3 months, 3 weeks ago

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