Question

diketahui fungsi f(x)=2x-3 dan (fog) (x) =2x²+4x+1. nilai g(-1) adalah..

pakai jalan​

Answer 1

• [tex](fog)(x) = {2x}^{2} + 4x + 1[/tex]
• [tex]f(g(x)) = {2x}^{2} + 4x + 1[/tex]
• [tex]2(g(x)) - 3 = {2x}^{2} + 4x + 1[/tex]
• [tex]2(g(x)) = {2x}^{2} + 4x + 1 - ( - 3)[/tex]
• [tex]2(g(x)) = {2x}^{2} + 4x + 1 + 3[/tex]
• [tex]2(g(x)) = {2x}^{2} + 4x + 4[/tex]
• [tex]g(x) = \frac{{2x}^{2} + 4x + 4}{2} [/tex]
• [tex]g(x) = {x}^{2} + 2x + 2[/tex]

Jadi, nilai dari g(-1) adalah

• [tex]g( - 1) = {( - 1)}^{2} + 2( - 1) + 2[/tex]
• [tex]g( - 1) = 1 + ( - 2) + 2[/tex]
• [tex]g( - 1) = ( - 1) + 2[/tex]
• [tex]g( - 1) = 1[/tex]

Answer 2

Diketahui fungsi f(x) = 2x – 3 dan (f o g)(x) = 2x² + 4x + 1.
Nilai g(–1) adalah 1.
 

Penjelasan dengan langkah-langkah

Fungsi Komposisi

Cara 1
Tentukan g(x) dan substitusi.

[tex]\begin{aligned}(f\circ g)(x)&=2x^2+4x+1\\2g(x)-3&=2x^2+4x+1\\2g(x)&=2x^2+4x+1+3\\&=2x^2+4x+4\\2g(x)&=2\left(x^2+2x+2\right)\\\implies g(x)&=x^2+2x+2\\\end{aligned}[/tex]

Maka:
g(–1) = (–1)² + 2(–1) + 2
⇔ g(–1) = 1 – 2 + 2
∴  g(–1) = 1
________

Cara 2

[tex]\begin{aligned}g(x)&=\left(\left(f^{-1}\circ f\right)\circ g\right)(x)\\&=\left(f^{-1}\circ\left(f\circ g\right)\right)(x)\\g(x)&=f^{-1}\left(\left(f\circ g\right)(x)\right)\end{aligned}[/tex]

Karena f(x) = 2x – 3, maka inversnya adalah:

[tex]\begin{aligned}f^{-1}(x)=\frac{x+3}{2}\end{aligned}[/tex]

Kita hitung (f o g)(–1).
(f o g)(–1) = 2·(–1)² + 4(–1) + 1
⇔ (f o g)(–1) = 2 – 4 + 1
⇔ (f o g)(–1) = –1

Maka:

[tex]\begin{aligned}g(-1)&=f^{-1}\left(\left(f\circ g\right)(-1)\right)\\&=f^{-1}(-1)\\&=\frac{-1+3}{2}=\frac{2}{2}\\\therefore\ g(-1)&=\bf1\end{aligned}[/tex]