Question

Pedro has created the function f(x) = the quantity of 4x minus 3, divided by 2 to represent the number of assignments he has completed, where x represents the number of weeks in the course. Pedro discovers that using the inverse function to solve for x = 30, he can predict when he will have 30 assignments completed. Explain to Pedro how to accomplish this, using complete sentences.

Answer 1

Let

x-------> the number of weeks in the course

f(x)------> the number of assignments that Pedro has completed

we know that

`f(x) = (4x-3)/2`

Step
`1`

Find the inverse function of f(x)

`f(x) = (4x-3)/2`

Let

y=f(x)

`y= (4x-3)/2`

exchange the variables x for y and y for x

`x= (4y-3)/2`

Clear variable y

`x= (4y-3)/2\\ 2x= 4y-3\\ 4y=2x+3\\ y=(2x+3)/4`

Let

`f(x)^(-1)=y`

`f(x)^(-1)=(2x+3)/4` ------> this is the inverse function

where

x ------> the number of assignments that Pedro has completed

f(x)^{-1}------> the number of weeks in the course

Solve for
`x=30`

substitute

`f(x)^(-1)=(2x+3)/4`

`f(x)^(-1)=(2*30+3)/4`

`f(x)^(-1)=15.75`

therefore

the answer is

`15(3)/(4)\ weeks`

Answer 2

Find when he will have 30 assignments completed for the inverse function.

To prove

As given

Pedro has created the function.

`f(x) = (4x -3)/(2)`

Where x represents the number of weeks in the course .

Pedro discovers that using the inverse function to solve for x = 30.

he can predict when he will have 30 assignments completed.

Now find the inverse of the function f(x) .

`Take\ y = f(x) = (4x -3)/(2)`

`2y + 3 = 4x`

`x = (2y + 3)/(4)`

Now change the y variable into x. (Because this is inverse function of f(x)).

Thus

`(f(x))^(-1) = (2x + 3)/(4)`

Thus this is the inverse function of f(x).

Put x = 30

`(f(x))^(-1) = (2* 30 + 3)/(4)`

`(f(x))^(-1) = (60 + 3)/(4)`

`(f(x))^(-1) = (63)/(4)`

`Therefore\ in\ (63)/(4)\ weeks\ Pedro\ completed\ 30\ assignment.`

In mixed fraction form

`Therefore\ in\ 15 (3)/(4)\ weeks\ Pedro\ completed\ 30\ assignment.`