f , g , h , k ( x ) f, g, h, k(x)

Algebra Level 3

The function f ( x ) = 1 1 + x f(x) = \frac{1}{1+x} is defined for all real values x ≠ − 1 , x\neq -1, and the functions f , g , and h f, g, \mbox{ and } h satisfy the equations f ( g ( x ) ) = x + 1 and h ( f ( x ) ) = x + 2. f(g(x)) = x+1 \mbox{ and } h(f(x)) = x + 2. If k ( x ) = g ( x ) + h ( x ) , k(x) = g(x) + h(x), then what is k ( 4 ) ? k(4)?

9 20 \frac{9}{20} 7 20 \frac{7}{20} 11 20 \frac{11}{20} 13 20 \frac{13}{20}

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Mahindra Jain by Mahindra Jain

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1 solution

Zakaria Sellami
May 8, 2014

f ( g ( 4 ) ) = 4 + 1 = 5 f(g(4))=4+1=5 then 1 1 + g ( 4 ) = 5 \frac{1}{1+g(4)}=5 then g ( 4 ) = − 4 / 5 g(4)=-4/5 . f ( x ) = 4 f(x)=4 means x = − 3 / 4 x=-3/4 . h ( f ( − 3 / 4 ) ) = − 3 / 4 + 2 = 5 / 4 h(f(-3/4))=-3/4+2=5/4 , hence, h ( 4 ) = 5 / 4 h(4)=5/4 .
k ( 4 ) = g ( 4 ) + h ( 4 ) = − 4 / 5 + 5 / 4 = 9 / 20 k(4)=g(4)+h(4)=-4/5+5/4=9/20

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