How Many f f 's Are Obsolete?

Algebra Level 1

Let f ( x ) = x + 1 x − 1 f(x) = \dfrac{x+1}{x-1} , where x ≠ 1 x \neq 1 . Find f ( f ( f ( f ( x ) ) ) ) f(f(f(f(x)))) .

x x x + 1 x+1 x − 1 x-1 x + 1 x − 1 \frac{x+1}{x-1}

Abc Xyz by abc xyz , 18, India

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2 solutions

Hung Woei Neoh
Jun 12, 2016

Relevant wiki: Function Composition

f ( f ( x ) ) = f ( x + 1 x − 1 ) = x + 1 x − 1 + 1 x + 1 x − 1 − 1 = x + 1 + 1 ( x − 1 ) x − 1 ÷ x + 1 − 1 ( x − 1 ) x − 1 = 2 x x − 1 × x − 1 2 = x f(f(x))\\ =f\left(\dfrac{x+1}{x-1}\right)\\ =\dfrac{\frac{x+1}{x-1} + 1}{\frac{x+1}{x-1} - 1}\\ =\dfrac{x+1 + 1(x-1)}{x-1} \div \dfrac{x+1-1(x-1)}{x-1}\\ =\dfrac{2x}{x-1} \times \dfrac{x-1}{2}\\ =x

Therefore,

f ( f ( f ( f ( x ) ) ) ) = f ( f ( x ) ) = x f(f(\color{#3D99F6}{f(f(x))}))\\ =f(f(x))\\ =\boxed{x}

8 Helpful 0 Interesting 0 Brilliant 0 Confused
Abc Xyz
Jun 12, 2016

Relevant wiki: Function Composition

f ( f ( f ( f ( x ) ) ) ) = f ( f ( f ( x + 1 x − 1 ) ) ) f(f(f(f(x)))) = f(f(f(\dfrac{x+1}{x-1})))

= f ( f ( x + 1 x − 1 + 1 x + 1 x − 1 − 1 ) ) = f(f(\dfrac{\frac{x+1}{x-1}+1}{\frac{x+1}{x-1}-1}))

= f ( f ( x ) ) = f(f(x))

= f ( x + 1 x − 1 ) = x + 1 x − 1 + 1 x + 1 x − 1 − 1 = x =f(\dfrac{x+1}{x-1}) = \dfrac{\frac{x+1}{x-1}+1}{\frac{x+1}{x-1}-1} = x

T h e r e f o r e Therefore

f ( f ( f ( f ( x ) ) ) ) = x f(f(f(f(x)))) = \boxed{x}

5 Helpful 0 Interesting 0 Brilliant 0 Confused

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