Diketahui dua fungsi f : R → R dan g : R → R dirumuskan oleh f(x) = 2x - 2 dan g(x) = x² + 4x - 3. Jika (gof)(b) = 2, maka nilai b adalah ³⁄₂ atau -³⁄₂
Pembahasan
- Menentukan (gof)(x)
(gof)(x) = g(f(x)
(gof)(x) = (2x - 2)² + 4(2x - 2) - 3
(gof)(x) = (4x² - 8x + 4) + (8x - 8) - 3
(gof)(x) = 4x² - 8x + 8x + 4 - 8 + 3
(gof)(x) = 4x² - 4 + 3
(gof)(x) = 4x² - 7
- Nilai b dari (gof)(b)
(gof)(x) = 4x² - 7
(gof)(b) = 4b² - 7
4b² - 7 = 2
4b² = 2 + 7
4b² = 9
b² = ⁹⁄₄
b = ±√⁹⁄₄
b = ±³⁄₂
b₁ = ³⁄₂ ⋁ b₂ = -³⁄₂
Jawab:
Penjelasan dengan langkah-langkah:
[tex](g\circ f)(b) = 2\\(g^{-1}\circ g\circ f)(b) = f(b) = g^{-1}(2)\\b = (f^{-1}\circ g^{-1})(2)\\g(x) = x^2+4x-3 = (x+2)^2-7\to g^{-1}(x) = -2 \pm \sqrt{x+7}\\f(x) = 2x-2 \to f^{-1}(x) = \dfrac{x+2}{2}\\g^{-1}(2) = -2 \pm 3 = -5\cup 1\\b = f^{-1}(-5) \cup f^{-1}(1)\\b = \dfrac{-5+2}{2}\cup \dfrac{1+2}{2}\\\\\boxed{\boxed{b = -\dfrac{3}{2} \cup \dfrac{3}{2}}}[/tex]
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