Diketahui fungsi - Universityku

f(x)=3x²+4x+1
g(x)=6x-3
h(x)=x-2/x+1

tentukan:
a. (fog)(x)
b. (hof)(3)
c. (hofog)(x)
d. h^-1(x)
e. (goh^-1)(x)
f. (fogoh^-1)(x)

Diketahui fungsi

Diketahui fungsi

$f(x) = 3 {x}^{2} + 4x + 1$

$g(x) = 6x - 3$

$h(x) = frac{x - 2}{x + 1}$

SOAL A.

$(fog)(x) = f(g(x))$

$(fog)(x) = f(6x - 3)$

$(fog)(x) = 3 {(6x - 3)}^{2} + 4(6x - 3) + 1$

$(fog)(x) =3(36 {x}^{2} - 36x + 9) + 24x - 12 + 1$

$(fog)(x) = 108 {x}^{2} - 108x + 27 + 24x - 12 + 1$

$(fog)(x) = 108 {x}^{2} - 108x + 24x + 27 - 12 + 1$

$(fog)(x) = 108 {x}^{2} - 84x + 16$

SOAL B.

$(hof)(3) = h(f(3))$

$(hof)(3) = h(3( {3}^{2} ) + 4(3) + 1)$

$(hof)(3) = h(27 + 12 + 1)$

$(hof)(3) = h(40)$

$(hof)(3) = frac{40 - 2}{40 + 1}$

$(hof)(3) = frac{38}{41}$

SOAL C.

$(hofog)(x) = h(fog(x))$

$(hofog)(x) = h(108 {x}^{2} - 84x + 16)$

$(hofog)(x) = frac{(108 {x}^{2} - 84x + 16) - 2 }{(108 {x}^{2} - 84x + 16) + 1 }$

$(hofog)(x) = frac{108 {x}^{2} - 84x + 16 - 2 }{108 {x}^{2} - 84x + 16 + 1 }$

$(hofog)(x) = frac{108 {x}^{2} - 84x + 14}{108 {x}^{2} - 84x + 17 }$

SOAL D.

$h(x) = frac{x - 2}{x + 1}$

Misalkan, $y = h(x).$

Maka, $y = frac{x - 2}{x + 1}$

$(y)(x + 1) = (x - 2)$

$xy + y = x - 2$

$xy - x = - y - 2$

$x(y - 1) = - y - 2$

$x = frac{ - y - 2}{y - 1}$

$x = frac{ - (y + 2)}{ - ( - y + 1)}$

$x = frac{y + 2}{1 - y}$

${h}^{ - 1} (x) = frac{x + 2}{1 - x}$

SOAL E.

$(go {h}^{ - 1} )(x) = g( {h}^{ - 1} (x))$

$(go {h}^{ - 1} )(x) = g( frac{x + 2}{1 - x})$

$(go {h}^{ - 1} )(x) = 6( frac{x + 2}{1 - x} ) - 3$

$(go {h}^{ - 1} )(x) = ( frac{6x + 12}{1 - x} ) - 3$

$(go {h}^{ - 1} )(x) = frac{6x + 12 - 3(1 - x)}{1 - x}$

$(go {h}^{ - 1} )(x) = frac{6x + 12 - 3 + x}{1 - x}$

$(go {h}^{ - 1} )(x) = frac{6x + x + 12 - 3}{1 - x}$

$(go {h}^{ - 1} )(x) = frac{7x + 9}{1 - x}$

SOAL F.

$(fogo {h}^{ - 1} )(x) = f(go {h}^{ - 1} (x))$

$(fogo {h}^{ - 1} )(x) = f( frac{7x + 9}{1 - x} )$

$(fogo {h}^{ - 1} )(x) = 3 {( frac{7x + 9}{1 - x}) }^{2} + 4( frac{7x + 9}{1 - x} ) + 1$

$(fogo {h}^{ - 1} )(x) = 3( frac{49 {x}^{2} + 126x + 81 }{ {(1 - x)}^{2} } ) + ( frac{28x + 36}{1 - x} ) + 1$

$(fogo {h}^{ - 1} )(x) = ( frac{147 {x}^{2} + 378x + 243}{ {(1 - x)}^{2} } ) + ( frac{(28x + 36)(1 - x)}{ {(1 - x)}^{2} } ) + frac{ {(1 - x)}^{2} }{ {(1 - x)}^{2} }$

$(fogo {h}^{ - 1} )(x) = frac{(147 {x}^{2} + 378x + 243) + ( - 28 {x}^{2} - 8x + 36) + ( {x}^{2} - 2x + 1) }{ {(1 - x)}^{2} }$

$(fogo {h}^{ - 1} )(x) = frac{147 {x}^{2} + 378x + 243 - 28 {x}^{2} - 8x + 36 + {x}^{2} - 2x + 1 }{ {(1 - x)}^{2} }$

$(fogo {h}^{ - 1} )(x) = frac{147 {x}^{2} - 28 {x}^{2} + {x}^{2} + 378x - 8x - 2x + 243 + 36 + 1 }{ {(1 - x)}^{2} }$

$(fogo {h}^{ - 1} )(x) = frac{120 {x}^{2} + 368x + 280}{ {(1 - x)}^{2} }$