Pertanyaan

Let f ( x ) = k x ( 1 − x ) and assume that k > 0 , show that the values of and b for which f ( a ) = b and f ( b ) = a are given by the formulas : a = 2 k 1 + k + ( k − 3 ) ( k + 1 ) and b = 2 k 1 + k − ( k − 3 ) ( k + 1 )

Let $f (x) = k x (1 - x)$ and assume that $k > 0$ , show that the values of $a$ and $b$ for which $f (a) = b$ and $f (b) = a$ are given by the formulas :

$a = \frac{1 + k + ( k - 3 ) ( k + 1 )}{2 k} and b = \frac{1 + k - ( k - 3 ) ( k + 1 )}{2 k}$

I. Sutiawan

Master Teacher

Mahasiswa/Alumni Universitas Pasundan

Jawaban terverifikasi

Jawaban

untuk dan berlaku

untuk $a equals fraction numerator 1 plus k plus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction$ dan $b equals fraction numerator 1 plus k minus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction$ berlaku f open parentheses a close parentheses equals b

f open parentheses a close parentheses equals b

f open parentheses b close parentheses equals a

Pembahasan

Diketahui dengan , dan . Akan kita tunjukan bahwa dan , maka: Jadi,untuk dan berlaku

Diketahui f open parentheses x close parentheses equals k x open parentheses 1 minus x close parentheses dengan k greater than 0 , $a equals fraction numerator 1 plus k plus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction$ dan $b equals fraction numerator 1 plus k minus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction$ . Akan kita tunjukan bahwa f open parentheses a close parentheses equals b dan f open parentheses b close parentheses equals a , maka:

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis a right parenthesis end cell equals cell k a left parenthesis 1 minus a right parenthesis end cell row blank equals cell k open parentheses fraction numerator 1 plus k plus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction close parentheses open parentheses 1 minus fraction numerator 1 plus k plus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction close parentheses end cell row blank equals cell k open parentheses fraction numerator 1 plus k plus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction close parentheses open parentheses fraction numerator 2 k minus open parentheses 1 plus k plus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root close parentheses over denominator 2 k end fraction close parentheses end cell row blank equals cell fraction numerator k plus k squared plus k square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction cross times fraction numerator k minus 1 minus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction end cell row blank equals cell fraction numerator 2 k open parentheses 1 plus k minus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root close parentheses over denominator 4 k squared end fraction end cell row blank equals cell fraction numerator 1 plus k minus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction end cell row blank equals cell b space left parenthesis terbukti right parenthesis end cell end table end style$

$begin mathsize 12px style table attributes columnalign right center left columnspacing 0px end attributes row cell f left parenthesis b right parenthesis end cell equals cell k b left parenthesis 1 minus b right parenthesis end cell row blank equals cell k open parentheses fraction numerator 1 plus k minus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction close parentheses open parentheses 1 minus fraction numerator 1 plus k minus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction close parentheses end cell row blank equals cell k open parentheses fraction numerator 1 plus k minus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction close parentheses open parentheses fraction numerator 2 k minus open parentheses 1 plus k minus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root close parentheses over denominator 2 k end fraction close parentheses end cell row blank equals cell fraction numerator k plus k squared minus k square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction cross times fraction numerator k minus 1 minus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction end cell row blank equals cell fraction numerator 2 k open parentheses 1 plus k plus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root close parentheses over denominator 4 k squared end fraction end cell row blank equals cell fraction numerator 1 plus k plus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction end cell row blank equals a row cell f left parenthesis b right parenthesis end cell equals cell a space left parenthesis terbukti right parenthesis end cell end table end style$

Jadi, untuk $a equals fraction numerator 1 plus k plus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction$ dan $b equals fraction numerator 1 plus k minus square root of open parentheses k minus 3 close parentheses open parentheses k plus 1 close parentheses end root over denominator 2 k end fraction$ berlaku

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