A Hard Function Problem

Based on 2013 Olympiad Problem

A function ff is defined for all xx and has the following property,

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

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

Note by Jason Chrysoprase
3 years, 4 months ago

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1 vote

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

If badc\dfrac{b}{a} \neq \dfrac{d}{c} , then the range is R{ac}\mathbb{R}-\{\frac{a}{c}\} if c0c \neq 0 and R\mathbb{R} otherwise.

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

In conclusion, the range is R{ac}\mathbb{R}-\{\frac{a}{c}\} if c0c \neq 0 otherwise R\mathbb{R}.


Edit:

After finding a,b,c,da,b, c, d,

Either f(x)=xf(x)=x or f(x)=58x19×97x58f(x)=\dfrac{58x-19 \times 97}{x-58}

So, the range is either R\mathbb{R} or R{58}\mathbb{R} - \{58\}

Deeparaj Bhat - 3 years, 4 months ago

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So every xx we put, the answer is always real ?

So how about f(0)f(0)?

Jason Chrysoprase - 3 years, 4 months ago

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Oops, sorry for late reply. Whats the problem if we put f(0)f(0).
f(0)=a×0+bc×0+d=bdf(0) = \dfrac{a×0 + b}{c×0 + d} = \dfrac{b}{d}.
Since dc\dfrac{-d}{c} is removed from its domain, the answer bd\dfrac{b}{d} is real too.

Ashish Siva - 3 years, 4 months ago

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I don't know functions well , can you please explain a bit more on how you took the range?

Rishabh Tiwari - 3 years, 4 months ago

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The answer is R58\mathbb{R} - {58} since there is no xx value for 5858

Jason Chrysoprase - 3 years, 4 months ago

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Domain

xR:x58{x\in \mathbb{R} : x \neq 58}

Range

fR:f58{f \in \mathbb{R} : f \neq 58}

Jason Chrysoprase - 3 years, 4 months ago

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Check this out

Calvin Lin Staff - 3 years, 4 months ago

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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 years, 4 months ago

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Practice the challenge quizzes in the chapter and read the wikis.

Calvin Lin Staff - 3 years, 4 months ago

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@Calvin Lin Thank you sir :-)

Rishabh Tiwari - 3 years, 4 months ago

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Wow, that guy really wrote the same problem. We both came from Indonesia :)

Jason Chrysoprase - 3 years, 4 months ago

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why not me???

Ayush G Rai - 3 years, 4 months ago

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Sorry, forgot u :(

Jason Chrysoprase - 3 years, 4 months ago

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