Nabila A
10 Februari 2023 15:17
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Nabila A
10 Februari 2023 15:17
Pertanyaan
Diketahui f(x) = 2x/3-x dan x≠3 Tentukan (fof^-1)(x)
Diketahui f(x) = 2x/3-x dan x≠3
Tentukan (fof^-1)(x)
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W. Lestari
Mahasiswa/Alumni Universitas Sriwijaya
11 Februari 2023 02:53
<p>Jawaban : (fof<sup>-1</sup>)(x) = x</p><p> </p><p>Ingat kembali:</p><p>(fog)(x) = f(g(x))</p><p>(gof)(x) = g(f(x))</p><p>f(x) = y → f<sup>-1</sup>(y) = x</p><p> </p><p>Diketahui: f(x) = (2x) / (3 - x)</p><p> </p><p>Maka:</p><p>f(x) = (2x) / (3 - x)</p><p>y = (2x) / (3 - x)</p><p>y(3 - x) = 2x</p><p>3y - xy = 2x</p><p>2x = 3y - xy</p><p>2x + xy = 3y</p><p>x(2 + y) = 3y</p><p>x = (3y) / (2 + y)</p><p> f<sup>-1</sup>(y) = (3y) / (2 + y)</p><p> f<sup>-1</sup>(x) = (3x) / (2 + x)</p><p> </p><p>Sehingga:</p><p>(fof<sup>-1</sup>)(x) = f(f<sup>-1</sup>(x))</p><p>(fof<sup>-1</sup>)(x) = f[(3x)/(2 + x)]</p><p>(fof<sup>-1</sup>)(x) = 2[(3x)/(2 + x)] / [3 - (3x)/(2 + x)]</p><p>(fof<sup>-1</sup>)(x) = [(6x)/(2 + x)] / [(3(2 + x) - 3x) / (2 + x)] </p><p>(fof<sup>-1</sup>)(x) = [(6x)/(2 + x)] / [(6)/(2 + x)]</p><p>(fof<sup>-1</sup>)(x) = [(6x)/(2 + x)] . [(2 + x)/(6)]</p><p>(fof<sup>-1</sup>)(x) = x</p><p> </p><p>Jadi, (fof<sup>-1</sup>)(x) = x.</p>
Jawaban : (fof-1)(x) = x
Ingat kembali:
(fog)(x) = f(g(x))
(gof)(x) = g(f(x))
f(x) = y → f-1(y) = x
Diketahui: f(x) = (2x) / (3 - x)
Maka:
f(x) = (2x) / (3 - x)
y = (2x) / (3 - x)
y(3 - x) = 2x
3y - xy = 2x
2x = 3y - xy
2x + xy = 3y
x(2 + y) = 3y
x = (3y) / (2 + y)
f-1(y) = (3y) / (2 + y)
f-1(x) = (3x) / (2 + x)
Sehingga:
(fof-1)(x) = f(f-1(x))
(fof-1)(x) = f[(3x)/(2 + x)]
(fof-1)(x) = 2[(3x)/(2 + x)] / [3 - (3x)/(2 + x)]
(fof-1)(x) = [(6x)/(2 + x)] / [(3(2 + x) - 3x) / (2 + x)]
(fof-1)(x) = [(6x)/(2 + x)] / [(6)/(2 + x)]
(fof-1)(x) = [(6x)/(2 + x)] . [(2 + x)/(6)]
(fof-1)(x) = x
Jadi, (fof-1)(x) = x.
· 5.0 (1)
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