Given that:
The function to represent the growth of Neveah hair is
[tex]f(x)=\dfrac{2x+4}{3}[/tex]
where, x represents the number of months since her last hair cut.
To find:
The inverse of function and time when her hair will have grown 6 inches.
Solution:
We have,
[tex]f(x)=\dfrac{2x+4}{3}[/tex]
Put f(x)=y.
[tex]y=\dfrac{2x+4}{3}[/tex]
Interchange x and y.
[tex]x=\dfrac{2y+4}{3}[/tex]
Isolate y.
[tex]3x=2y+4[/tex]
[tex]3x-4=2y[/tex]
[tex]\dfrac{3x-4}{2}=y[/tex]
Put [tex]y=f^{-1}(x)[/tex].
[tex]f^{-1}(x)=\dfrac{3x-4}{2}[/tex]
So, the required inverse function is [tex]f^{-1}(x)=\dfrac{3x-4}{2}[/tex]. It represents the number of months to grow her hair x inches.
She can predict when her hair will have grown 6 inches by putting x=6 in the inverse function.
Put x=6 in the inverse function.
[tex]f^{-1}(6)=\dfrac{3(6)-4}{2}[/tex]
[tex]f^{-1}(6)=\dfrac{18-4}{2}[/tex]
[tex]f^{-1}(6)=\dfrac{14}{2}[/tex]
[tex]f^{-1}(6)=7[/tex]
Therefore, growth of her hair is 6 inches in 7 months.
