Question

Neveah has created the function f(x) =the quantity of 2x plus 4, divided by 3 to represent the growth of her hair, where x represents the number of months since her last hair cut. Neveah discovers that, using the inverse function to solve for x = 6, she can predict when her hair will have grown 6 inches. Explain to Neveah how to accomplish this, using complete sentences. (10 points)

Answer 1

Given that:

The function to represent the growth of Neveah hair is

$f(x)=\dfrac{2x+4}{3}$

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,

$f(x)=\dfrac{2x+4}{3}$

Put f(x)=y.

$y=\dfrac{2x+4}{3}$

Interchange x and y.

$x=\dfrac{2y+4}{3}$

Isolate y.

$3x=2y+4$

$3x-4=2y$

$\dfrac{3x-4}{2}=y$

Put $y=f^{-1}(x)$.

$f^{-1}(x)=\dfrac{3x-4}{2}$

So, the required inverse function is $f^{-1}(x)=\dfrac{3x-4}{2}$. 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.

$f^{-1}(6)=\dfrac{3(6)-4}{2}$

$f^{-1}(6)=\dfrac{18-4}{2}$

$f^{-1}(6)=\dfrac{14}{2}$

$f^{-1}(6)=7$

Therefore, growth of her hair is 6 inches in 7 months.