Pertanyaan

Diketahui fungsi f ( 2 x 1 ) = x + 4 dan g ( x ) = x − 3 x + 1 , x  = 3 . Rumus komposisi fungsi ( g ∘ f ) ( x ) adalah ....

Diketahui fungsi $f (\frac{1}{2 x}) = x + 4$ dan $g (x) = \frac{x + 1}{x - 3}$ , $x \neq = 3$ . Rumus komposisi fungsi $(g \circ f) (x)$ adalah ....

S. Nur

Master Teacher

Jawaban terverifikasi

Jawaban

.

$begin mathsize 14px style open parentheses g ring operator f close parentheses open parentheses x close parentheses equals fraction numerator 10 x plus 1 over denominator 2 x plus 1 end fraction end style$ .

Pembahasan

Diketahui , misalkan maka diperoleh sehingga dengan demikian komposisi fungsi adalah Jadi, .

Diketahui $begin mathsize 14px style f open parentheses fraction numerator 1 over denominator 2 x end fraction close parentheses equals x plus 4 end style$ , misalkan $begin mathsize 14px style y equals fraction numerator 1 over denominator 2 x end fraction end style$ maka diperoleh $begin mathsize 14px style x equals fraction numerator 1 over denominator 2 y end fraction end style$ sehingga

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses y close parentheses end cell equals cell fraction numerator 1 over denominator 2 y end fraction plus 4 end cell row blank equals cell fraction numerator 1 plus 8 y over denominator 2 y end fraction end cell row cell f open parentheses x close parentheses end cell equals cell fraction numerator 1 plus 8 x over denominator 2 x end fraction end cell row blank blank blank end table end style$

dengan demikian komposisi fungsi adalah

$begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses open parentheses x close parentheses end cell equals cell g open parentheses f open parentheses x close parentheses close parentheses end cell row blank equals cell g open parentheses fraction numerator 1 plus 8 x over denominator 2 x end fraction close parentheses end cell row blank equals cell fraction numerator open parentheses fraction numerator 1 plus 8 x over denominator 2 x end fraction close parentheses plus 1 over denominator open parentheses fraction numerator 1 plus 8 x over denominator 2 x end fraction close parentheses minus 3 end fraction end cell row blank equals cell fraction numerator fraction numerator 1 plus 8 x plus 2 x over denominator 2 x end fraction over denominator fraction numerator 1 plus 8 x minus 6 x over denominator 2 x end fraction end fraction end cell row blank equals cell fraction numerator 10 x plus 1 over denominator 2 x plus 1 end fraction end cell end table end style$

Jadi, $begin mathsize 14px style open parentheses g ring operator f close parentheses open parentheses x close parentheses equals fraction numerator 10 x plus 1 over denominator 2 x plus 1 end fraction end style$ .