Bantuin kawab soal ka

Bantuin kawab soal ka

$displaystyle f(x) = frac{4-x}{x-1}, : xneq 1 \ g(x) = frac{x}{2x+1}, : xneq -frac{1}{2} \ big(fcirc g^{-1}big)^{-1}(x) = dots?$

Penyelesaian:

$displaystyle g(x) = frac{x}{2x+1} \ y = frac{x}{2x+1} \ 2yx+y = x \ x-2yx = y \ x(1-2y) = y \ x = frac{y}{1-2y} \ therefore g^{-1}(x) = frac{x}{1-2x}$

$displaystyle begin{aligned} big(fcirc g^{-1}big)(x) &= f[ g(x) ] \ &= frac{4-[g(x)]}{[g(x)]-1} \ &= frac{4-left(frac{x}{1-2x}right)}{left(frac{x}{1-2x}right)-1} \ &= frac{frac{4(1-2x)-x}{1-2x}}{frac{x-(1-2x)}{1-2x}} \ &= frac{4-9x}{3x-1}end{aligned} \ therefore big(fcirc g^{-1}big)(x) = frac{4-9x}{3x-1}$

$displaystyle big(fcirc g^{-1}big)(x) = frac{4-9x}{3x-1} \ y = frac{4-9x}{3x-1} \ 3yx-y = 4-9x \ 3yx+9x = 4+y \ x(3y+9) = 4+y \ x = frac{4+y}{3y+9} \ therefore big(fcirc g^{-1}big)^{-1}(x) = frac{4+x}{3x+9}$

Jawaban:

$displaystyle boxed{bold{B. : frac{x+4}{3x+9} }}$

Jawab:

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