Given:
[tex]\begin{gathered} F(x)=\sqrt{x-5}+4 \\ G(x)=(x-4)^2+5 \end{gathered}[/tex]
Required:
Find F(x) and G(x) are inverse functions or not.
Explanation:
Given that
[tex]\begin{gathered} F(x)=\sqrt{x-5}+4 \\ G(x)=(x-4)^{2}+5 \end{gathered}[/tex]
Let
[tex]F(x)=y[/tex][tex]\begin{gathered} y=\sqrt{x-5}+4 \\ y-4=\sqrt{x-5} \end{gathered}[/tex]
Take the square on both sides.
[tex](y-4)^2=x-5[/tex]
Interchange x and y as:
[tex]\begin{gathered} (x-4)^2=y-5 \\ y=(x-4)^2+5 \end{gathered}[/tex]
Substitute y = G(x)
[tex]G(x)=(x-4)^2+5[/tex]
This is the G(x) function.
So F(x) and G(x) are inverse functions.
[tex]\begin{gathered} G(x)-5=(x-4)^2 \\ \sqrt{G(x)-5}=x-4 \\ x=\sqrt{G(x)-5}+4 \end{gathered}[/tex]
Final Answer:
Option A is the correct answer.
