Beautiful Symmetric Rational Function

Algebra Level 3

f ( x ) = ( x − 1 ) ( x − 2 ) ( x − 3 ) ( x − 4 ) ( x − 3 ) ( x − 2 ) ( x − 1 ) ( x − 2 ) ( x − 4 ) ( x − 2 ) f(x) = \dfrac{(x-1)(x-2)(x-3)(x-4)(x-3)(x-2)(x-1)}{(x-2)(x-4)(x-2)}

How many values of x x satisfy the equation f ( x ) = 1 f(x) = 1 ?

0 1 3 More than 4 2 4

Pranshu Gaba by Pranshu Gaba , 22, India

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2 solutions

Rishabh Jain
Feb 12, 2016

x ≠ 2 , 4 x\neq2,4 since for these values f(x) is undefined.
Hence f ( x ) = ( x − 1 ) 2 ( x − 3 ) 2 f(x)=(x-1)^2(x-3)^2 .
For f ( x ) = 1 f(x)=1 , we have ( x − 1 ) ( x − 3 ) = ± 1 (x-1)(x-3)=\pm1
1. ( x − 1 ) ( x − 3 ) = 1 ⇒ x 2 − 4 x + 2 = 0 ⇒ x = 2 ± 2 1. (x-1)(x-3)=1\Rightarrow x^2-4x+2=0\Rightarrow x=2\pm \sqrt2 2. ( x − 1 ) ( x − 3 ) = − 1 ⇒ x 2 − 4 x + 4 = 0 ⇒ x = 2 2. (x-1)(x-3)=-1\Rightarrow x^2-4x+4=0\Rightarrow x=2 Since already stated x ≠ 2 x\neq2 , hence we get x = 2 ± 2 x=2\pm \sqrt2 i.e 2 solutions .

13 Helpful 0 Interesting 0 Brilliant 0 Confused

Please post a graphical solution for this question

Azmat Arshi - 5 years, 3 months ago

Answer is 2±√2

0 Helpful 0 Interesting 1 Brilliant 0 Confused

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