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)
[tex]f(x) = 3 {x}^{2} + 4x + 1[/tex]
[tex]g(x) = 6x - 3[/tex]
[tex]h(x) = \frac{x - 2}{x + 1} [/tex]
SOAL A.
[tex](fog)(x) = f(g(x))[/tex]
[tex](fog)(x) = f(6x - 3)[/tex]
[tex](fog)(x) = 3 {(6x - 3)}^{2} + 4(6x - 3) + 1[/tex]
[tex](fog)(x) =3(36 {x}^{2} - 36x + 9) + 24x - 12 + 1[/tex]
[tex](fog)(x) = 108 {x}^{2} - 108x + 27 + 24x - 12 + 1[/tex]
[tex](fog)(x) = 108 {x}^{2} - 108x + 24x + 27 - 12 + 1[/tex]
[tex](fog)(x) = 108 {x}^{2} - 84x + 16[/tex]
SOAL B.
[tex](hof)(3) = h(f(3))[/tex]
[tex](hof)(3) = h(3( {3}^{2} ) + 4(3) + 1)[/tex]
[tex](hof)(3) = h(27 + 12 + 1)[/tex]
[tex](hof)(3) = h(40)[/tex]
[tex](hof)(3) = \frac{40 - 2}{40 + 1} [/tex]
[tex](hof)(3) = \frac{38}{41} [/tex]
SOAL C.
[tex](hofog)(x) = h(fog(x))[/tex]
[tex](hofog)(x) = h(108 {x}^{2} - 84x + 16)[/tex]
[tex](hofog)(x) = \frac{(108 {x}^{2} - 84x + 16) - 2 }{(108 {x}^{2} - 84x + 16) + 1 } [/tex]
[tex](hofog)(x) = \frac{108 {x}^{2} - 84x + 16 - 2 }{108 {x}^{2} - 84x + 16 + 1 } [/tex]
[tex](hofog)(x) = \frac{108 {x}^{2} - 84x + 14}{108 {x}^{2} - 84x + 17 } [/tex]
SOAL D.
[tex]h(x) = \frac{x - 2}{x + 1} [/tex]
Misalkan, [tex]y = h(x).[/tex]
Maka, [tex]y = \frac{x - 2}{x + 1} [/tex]
[tex](y)(x + 1) = (x - 2)[/tex]
[tex]xy + y = x - 2[/tex]
[tex]xy - x = - y - 2[/tex]
[tex]x(y - 1) = - y - 2[/tex]
[tex]x = \frac{ - y - 2}{y - 1} [/tex]
[tex]x = \frac{ - (y + 2)}{ - ( - y + 1)} [/tex]
[tex]x = \frac{y + 2}{1 - y} [/tex]
[tex] {h}^{ - 1} (x) = \frac{x + 2}{1 - x} [/tex]
SOAL E.
[tex](go {h}^{ - 1} )(x) = g( {h}^{ - 1} (x))[/tex]
[tex](go {h}^{ - 1} )(x) = g( \frac{x + 2}{1 - x}) [/tex]
[tex](go {h}^{ - 1} )(x) = 6( \frac{x + 2}{1 - x} ) - 3[/tex]
[tex](go {h}^{ - 1} )(x) = ( \frac{6x + 12}{1 - x} ) - 3[/tex]
[tex](go {h}^{ - 1} )(x) = \frac{6x + 12 - 3(1 - x)}{1 - x} [/tex]
[tex](go {h}^{ - 1} )(x) = \frac{6x + 12 - 3 + x}{1 - x} [/tex]
[tex](go {h}^{ - 1} )(x) = \frac{6x + x + 12 - 3}{1 - x} [/tex]
[tex](go {h}^{ - 1} )(x) = \frac{7x + 9}{1 - x} [/tex]
SOAL F.
[tex](fogo {h}^{ - 1} )(x) = f(go {h}^{ - 1} (x))[/tex]
[tex](fogo {h}^{ - 1} )(x) = f( \frac{7x + 9}{1 - x} )[/tex]
[tex](fogo {h}^{ - 1} )(x) = 3 {( \frac{7x + 9}{1 - x}) }^{2} + 4( \frac{7x + 9}{1 - x} ) + 1[/tex]
[tex](fogo {h}^{ - 1} )(x) = 3( \frac{49 {x}^{2} + 126x + 81 }{ {(1 - x)}^{2} } ) + ( \frac{28x + 36}{1 - x} ) + 1[/tex]
[tex](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} } [/tex]
[tex](fogo {h}^{ - 1} )(x) = \frac{(147 {x}^{2} + 378x + 243) + ( - 28 {x}^{2} - 8x + 36) + ( {x}^{2} - 2x + 1) }{ {(1 - x)}^{2} } [/tex]
[tex](fogo {h}^{ - 1} )(x) = \frac{147 {x}^{2} + 378x + 243 - 28 {x}^{2} - 8x + 36 + {x}^{2} - 2x + 1 }{ {(1 - x)}^{2} } [/tex]
[tex](fogo {h}^{ - 1} )(x) = \frac{147 {x}^{2} - 28 {x}^{2} + {x}^{2} + 378x - 8x - 2x + 243 + 36 + 1 }{ {(1 - x)}^{2} } [/tex]
[tex](fogo {h}^{ - 1} )(x) = \frac{120 {x}^{2} + 368x + 280}{ {(1 - x)}^{2} } [/tex]
[answer.2.content]
