Diketahui f(x)=ax+a−x  dengan a>0. Buktikan bahwa f(x+1)+f(x−1)=(a+a1​)f(x)

Pertanyaan

Diketahui  dengan . Buktikan bahwa

   

  1. ... 

  2. ... 

F. Ayudhita

Master Teacher

Jawaban terverifikasi

Jawaban

terbukti bahwa begin mathsize 14px style bold italic f begin bold style left parenthesis x plus 1 right parenthesis end style bold plus bold italic f begin bold style left parenthesis x minus 1 right parenthesis end style bold equals begin bold style left parenthesis a plus 1 over a right parenthesis end style bold italic f begin bold style left parenthesis x right parenthesis end style end style 

Pembahasan

Pembahasan

Ingat!

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell a to the power of negative m end exponent end cell equals cell 1 over a to the power of m end cell row cell a to the power of m plus n end exponent end cell equals cell a to the power of m times a to the power of n end cell row blank blank blank end table end style 

Akan dibuktikan begin mathsize 14px style f open parentheses x plus 1 close parentheses plus f open parentheses x minus 1 close parentheses equals open parentheses a plus 1 over a close parentheses f open parentheses x close parentheses end style 

table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell a to the power of x plus a to the power of negative x end exponent end cell row cell f open parentheses x plus 1 close parentheses end cell equals cell a to the power of x plus 1 end exponent plus a to the power of negative open parentheses x plus 1 close parentheses end exponent end cell row blank equals cell a to the power of x plus 1 end exponent plus a to the power of negative x minus 1 end exponent end cell row blank equals cell a to the power of x times a plus a to the power of negative x end exponent times a to the power of negative 1 end exponent end cell row cell f open parentheses x minus 1 close parentheses end cell equals cell a to the power of x minus 1 end exponent plus a to the power of negative open parentheses x minus 1 close parentheses end exponent end cell row blank equals cell a to the power of x minus 1 end exponent plus a to the power of negative x plus 1 end exponent end cell row blank equals cell a to the power of x times a to the power of negative 1 end exponent plus a to the power of negative x end exponent times a end cell end table  

Sehingga

 begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x plus 1 close parentheses plus f open parentheses x minus 1 close parentheses end cell equals cell a to the power of x times a plus a to the power of negative x end exponent times a to the power of negative 1 end exponent plus a to the power of x times a to the power of negative 1 end exponent plus a to the power of negative x end exponent times a end cell row blank equals cell a times a to the power of x plus a times a to the power of negative x end exponent plus a to the power of negative 1 end exponent times a to the power of x plus a to the power of negative 1 end exponent times a to the power of negative x end exponent end cell row blank equals cell open parentheses a plus a to the power of negative 1 end exponent close parentheses open parentheses a to the power of x plus a to the power of negative x end exponent close parentheses end cell row blank equals cell open parentheses a plus 1 over a close parentheses f open parentheses x close parentheses end cell end table end style   

Jadi, terbukti bahwa begin mathsize 14px style bold italic f begin bold style left parenthesis x plus 1 right parenthesis end style bold plus bold italic f begin bold style left parenthesis x minus 1 right parenthesis end style bold equals begin bold style left parenthesis a plus 1 over a right parenthesis end style bold italic f begin bold style left parenthesis x right parenthesis end style end style 

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