Jika f(x)=3x−4 dan g(x)=3−4x. Tentukanlah (g∘f)−1(x) dan (f∘g)−1(x).

Pertanyaan

Jika $f\left(x\right)=3x-4$ dan $g\left(x\right)=3-4x$ . Tentukanlah $\left(g\circ f\right)^{-1}\left(x\right)$ dan $\left(f\circ g\right)^{-1}\left(x\right).$

H. Nur

Master Teacher

Mahasiswa/Alumni Universitas Jember

Jawaban terverifikasi

Jawaban

bentuk dari ${\boldsymbol(\boldsymbol g\boldsymbol\circ\boldsymbol f\boldsymbol)}^{\boldsymbol-\mathbf1}{\boldsymbol(\boldsymbol x\boldsymbol)}\boldsymbol=$ $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator bold minus bold italic x bold minus bold 13 over denominator bold 12 end fraction end cell end table$ dan ${\boldsymbol(\boldsymbol f\boldsymbol\circ\boldsymbol g\boldsymbol)}^{\boldsymbol-\mathbf1}{\boldsymbol(\boldsymbol x\boldsymbol)}\boldsymbol=$ $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator bold minus bold italic x bold plus bold 5 over denominator bold 12 end fraction end cell end table$ .

Pembahasan

Jawaban yang benar untuk pertanyaan tersebut adalah ${\boldsymbol(\boldsymbol g\boldsymbol\circ\boldsymbol f\boldsymbol)}^{\boldsymbol-\mathbf1}{\boldsymbol(\boldsymbol x\boldsymbol)}\boldsymbol=$ $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator bold minus bold italic x bold minus bold 13 over denominator bold 12 end fraction end cell end table$ dan ${\boldsymbol(\boldsymbol f\boldsymbol\circ\boldsymbol g\boldsymbol)}^{\boldsymbol-\mathbf1}{\boldsymbol(\boldsymbol x\boldsymbol)}\boldsymbol=$ $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator bold minus bold italic x bold plus bold 5 over denominator bold 12 end fraction end cell end table$ .

Ingat!

Ada 2 cara menentukan formula invers fungsi komposisi, yaitu:

Mula-mula menentukan fungsi komposisinya, kemudian diinverskan.
Mula-mula menentukan invers masing-masing fungsi kemudian dikomposisikan.

Ingat pula $\left(g\circ f\right)^{-1}\left(x\right)=\left(f^{-1}\circ g^{-1}\right)\left(x\right)$ dan $\left(f\circ g\right)^{-1}\left(x\right)=\left(g^{-1}\circ f^{-1}\right)\left(x\right)$ .

Diketahui:

$f\left(x\right)=3x-4\\g\left(x\right)=3-4x$

Ditanya:

$\left(g\circ f\right)^{-1}\left(x\right)$ dan $\left(f\circ g\right)^{-1}\left(x\right)$

Jawab:

Cari $f^{-1}\left(x\right)$ :

$table attributes columnalign right center left columnspacing 0px end attributes row cell f open parentheses x close parentheses end cell equals cell 3 x minus 4 end cell row y equals cell 3 x minus 4 end cell row cell 3 x end cell equals cell y plus 4 end cell row x equals cell fraction numerator y plus 4 over denominator 3 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses y close parentheses end cell equals cell fraction numerator y plus 4 over denominator 3 end fraction end cell row cell f to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator x plus 4 over denominator 3 end fraction end cell end table$

Cari $g^{-1}\left(x\right)$ :

$table attributes columnalign right center left columnspacing 0px end attributes row cell g open parentheses x close parentheses end cell equals cell 3 minus 4 x end cell row y equals cell 3 minus 4 x end cell row cell 4 x end cell equals cell 3 minus y end cell row x equals cell fraction numerator 3 minus y over denominator 4 end fraction end cell row cell g to the power of negative 1 end exponent open parentheses y close parentheses end cell equals cell fraction numerator 3 minus y over denominator 4 end fraction end cell row cell g to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell fraction numerator 3 minus x over denominator 4 end fraction end cell end table$

Mencari $\left(g\circ f\right)^{-1}\left(x\right)$ dengan sifat invers $\left(g\circ f\right)^{-1}\left(x\right)=\left(f^{-1}\circ g^{-1}\right)\left(x\right)$ :

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses g ring operator f close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell open parentheses f to the power of negative 1 end exponent ring operator g to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell row blank equals cell f to the power of negative 1 end exponent open parentheses g to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell row blank equals cell f to the power of negative 1 end exponent open parentheses fraction numerator 3 minus x over denominator 4 end fraction close parentheses end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator 3 minus x over denominator 4 end fraction end style close parentheses plus 4 over denominator 3 end fraction end cell row blank equals cell fraction numerator open parentheses begin display style fraction numerator 3 minus x over denominator 4 end fraction end style close parentheses minus begin display style 16 over 4 end style over denominator 3 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 3 minus x minus 16 over denominator 4 end fraction end style over denominator 3 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator negative x minus 13 over denominator 4 end fraction end style over denominator 3 end fraction end cell row blank equals cell fraction numerator negative x minus 13 over denominator 4 end fraction times 1 third end cell row blank equals cell fraction numerator negative x minus 13 over denominator 12 end fraction end cell end table$

Mencari $\left(f\circ g\right)^{-1}\left(x\right)$ dengan sifat invers $\left(f\circ g\right)^{-1}\left(x\right)=\left(g^{-1}\circ f^{-1}\right)\left(x\right)$ :

$table attributes columnalign right center left columnspacing 0px end attributes row cell open parentheses f ring operator g close parentheses to the power of negative 1 end exponent open parentheses x close parentheses end cell equals cell open parentheses g to the power of negative 1 end exponent ring operator f to the power of negative 1 end exponent close parentheses open parentheses x close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses f to the power of negative 1 end exponent open parentheses x close parentheses close parentheses end cell row blank equals cell g to the power of negative 1 end exponent open parentheses fraction numerator x plus 4 over denominator 3 end fraction close parentheses end cell row blank equals cell fraction numerator 3 minus open parentheses begin display style fraction numerator x plus 4 over denominator 3 end fraction end style close parentheses over denominator 4 end fraction end cell row blank equals cell fraction numerator begin display style 9 over 3 end style minus open parentheses begin display style fraction numerator x plus 4 over denominator 3 end fraction end style close parentheses over denominator 4 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator 9 minus x minus 4 over denominator 3 end fraction end style over denominator 4 end fraction end cell row blank equals cell fraction numerator begin display style fraction numerator negative x plus 5 over denominator 3 end fraction end style over denominator 4 end fraction end cell row blank equals cell fraction numerator negative x plus 5 over denominator 3 end fraction times 1 fourth end cell row blank equals cell fraction numerator negative x plus 5 over denominator 12 end fraction end cell end table$

Dengan demikian, bentuk dari ${\boldsymbol(\boldsymbol g\boldsymbol\circ\boldsymbol f\boldsymbol)}^{\boldsymbol-\mathbf1}{\boldsymbol(\boldsymbol x\boldsymbol)}\boldsymbol=$ $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator bold minus bold italic x bold minus bold 13 over denominator bold 12 end fraction end cell end table$ dan ${\boldsymbol(\boldsymbol f\boldsymbol\circ\boldsymbol g\boldsymbol)}^{\boldsymbol-\mathbf1}{\boldsymbol(\boldsymbol x\boldsymbol)}\boldsymbol=$ $table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell fraction numerator bold minus bold italic x bold plus bold 5 over denominator bold 12 end fraction end cell end table$ .