Pertanyaan

Diketahui fungsi f : R → R , g : R → R , dan h : R → R . Jika fungsi f ( x ) = 4 − 3 x 2 , g ( x ) = x x + 2 , dan ( f ∘ g ∘ h ) − 1 ( x ) = 3 x . Tentukan fungsi h ( x ) .

Diketahui fungsi $f : R \to R, g : R \to R$ , dan $h : R \to R .$ Jika fungsi $f (x) = 4 - 3 x^{2}, g (x) = \frac{x + 2}{x},$ dan $(f \circ g \circ h)^{- 1} (x) = 3 x .$ Tentukan fungsi $h (x) .$

H. Firmansyah

Master Teacher

Mahasiswa/Alumni Universitas Gadjah Mada

Jawaban terverifikasi

Jawaban

diperoleh nilai h ( x ) = ( x 2 − 12 + 9 ) − 18 .

diperoleh nilai

h (x) = \frac{- 18}{( x ^{2} - 12 + 9 )}

Pembahasan

Diketahui ( f ∘ g ∘ h ) − 1 ( x ) = 3 x dan y = 3 x maka y 2 = 3 x ⇒ x = 3 y 2 Lebih lanjut, diperoleh ( f ∘ g ∘ h ) ( x ) f ( g ( h ( x ))) f ( h ( x ) h ( x ) + 2 ) 4 − 3 ( h ( x ) h ( x ) + 2 ) 2 4 − h ( x ) 3 h ( x ) + 6 h ( x ) 3 h ( x ) + 6 h ( x ) 3 h ( x ) + 6 9 h ( x ) + 18 ( x 2 − 12 + 9 ) h ( x ) + 18 ( x 2 − 12 + 9 ) h ( x ) h ( x ) = = = = = = = = = = = 3 x 2 3 x 2 3 x 2 3 x 2 3 x 2 4 − 3 x 2 3 12 − x 2 12 h ( x ) − x 2 h ( x ) 0 − 18 ( x 2 − 12 + 9 ) − 18 Dengan demikian, diperoleh nilai h ( x ) = ( x 2 − 12 + 9 ) − 18 .

Diketahui

$(f \circ g \circ h)^{- 1} (x) = 3 x dan y = 3 x maka y^{2} = 3 x \Rightarrow x = \frac{y ^{2}}{3}$

Lebih lanjut, diperoleh

$(f \circ g \circ h) (x) f (g (h (x))) f (\frac{h ( x ) + 2}{h ( x )}) 4 - 3 (\frac{h ( x ) + 2}{h ( x )})^{2} 4 - \frac{3 h ( x ) + 6}{h ( x )} \frac{3 h ( x ) + 6}{h ( x )} \frac{3 h ( x ) + 6}{h ( x )} 9 h (x) + 18 (x^{2} - 12 + 9) h (x) + 18 (x^{2} - 12 + 9) h (x) h (x) = = = = = = = = = = = \frac{x ^{2}}{3} \frac{x ^{2}}{3} \frac{x ^{2}}{3} \frac{x ^{2}}{3} \frac{x ^{2}}{3} 4 - \frac{x ^{2}}{3} \frac{12 - x ^{2}}{3} 12 h (x) - x^{2} h (x) 0 - 18 \frac{- 18}{( x ^{2} - 12 + 9 )}$