Linear Transformation

Algebra Level 2

Let f f be a fractional linear transformation such that f ( 2 ) = 3 , f ( 5 ) = 5 , f(2)=3, f(5)=5, and f ( 7 ) = 6 1 3 f(7)=6 \frac 13 . What is f ( 8 ) f(8) ?


The answer is 7.

A P by A P , 26, USA

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1 solution

Hung Woei Neoh
Jun 7, 2016

We let f ( x ) = a x + b f(x) = ax+b . This gives:

f ( 2 ) = 3 ⟹ 2 a + b = 3 f ( 5 ) = 5 ⟹ 5 a + b = 5 f ( 5 ) − f ( 2 ) = 5 − 3 ( 5 a + b ) − ( 2 a + b ) = 2 3 a = 2 a = 2 3 f(2) = 3 \implies 2a+b = 3\\ f(5) = 5 \implies 5a+b = 5\\ f(5) - f(2) = 5-3\\ (5a+b) - (2a+b) = 2\\ 3a=2\\ a=\dfrac{2}{3}

Substitute to find b b :

2 ( 2 3 ) + b = 3 b = 3 − 4 3 = 5 3 2\left(\dfrac{2}{3}\right) + b = 3\\ b = 3 - \dfrac{4}{3} = \dfrac{5}{3}

Therefore, f ( x ) = 2 3 x + 5 3 f(x) = \dfrac{2}{3} x + \dfrac{5}{3}

⟹ f ( 8 ) = 2 3 ( 8 ) + 5 3 = 16 3 + 5 3 = 21 3 = 7 \implies f(8) = \dfrac{2}{3}(8) + \dfrac{5}{3} = \dfrac{16}{3} + \dfrac{5}{3} = \dfrac{21}{3} = \boxed{7}

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