# Find a formula for the inverse of the function f=(4x-1)/(2x+3)

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

To find the inverse function f(x)^-1 we will assume:

y=(4x-1)/(2x+3)

==> multiply by 2x+3

==> y(2x+3)= 4x-1

==> 2yx+3y=4x-1

==> 3y+1= 4x-2yx

==> 3y+1= x(4-2y)

==> x= (3y+1)/(4-2y)

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

We'll start to write:f(x)=(4x-1)/(2x+3) as y=(4x-1)/(2x+3)

Now, we'll solve this equation for x, multiplying y by (2x+3):

4x-1 = y(2x+3)

We'll open the brackets:

4x-1 = 2x*y + 3y

We'll move all terms containing x, to the left side:

4x-2x*y=3y+1

We'll factorize:

x(4-2y)=3y+1

x=(3y+1)/(4-2y)

Now, we'll interchange x and y:

y=(3x+1)/(4-2x)

So, the inverse function is:

**[f(x)]^(-1) = (3x+1)/(4-2x)**

Any function f(x) and its inverse function `f^-1(x)` follow the relation `f(f^-1(x)) = x` .

For the function `f(x)=(4x-1)/(2x+3)` , to determine the inverse, use the relation provided earlier. This gives:

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

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

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

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

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

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

The required inverse of the function `f(x) = (4x-1)/(2x+3)` is` f^-1(x) = (3x + 1)/(4 - 2x)`

To find a formula for finding the inverse of f(x) = (4x-1)/(2x+3).

Solution:

Let y = (4x-1)/(2x+3).....(1). We shall try to make x as the subject instead of y and then interchange x and y:

Multiplying both sides of (1) by (2x+3):

y(2x+3) = 4x-1.Or

2yx-3y = 4x-1. Or

2yx-4x = -1+3y. Or

2x(y-2) = 3y-1. Dividing both sides by (y-2),

x = (3y-1)/(y-2).

Interchanging xand y,

y = (3x-1)/(x-2) is the inverse of the given function.