Find the equation

Question

Find the equation for f^-1(x) given f(x)=3x-2 OVER 4x-5

Answer 1

There is a formula for it, but here goes

y = (3x-2)/(4x-5)

swaps:

x = (3y-2)/(4y-5)

x(4y-5) = 3y-2

4xy - 5x = 3y - 2

4xy - 3y = 5x-2

y(4x-3) = 5x-2

y = (5x-2)/(4x-3) = f_inv

=============================

check:

3(5x-2)/(4x-3) - 2

-------------------------- <--- f comp f_inv

4(5x-2)/(4x-3) - 5

multiplies top and bottom by (4x-3):

3(5x-2) - 2(4x-3)

----------------------

4(5x-2) - 5(4x-3)

15x-6 - 8x + 6

-------------------

20x-8 - 20x + 15

7x

-------- =

7

x

check:

5 (3x-2)/(4x-5) - 2

-------------------- <-- f_inv comp f

4 (3x-2)/(4x-5) - 3

5(3x-2) - 2(4x-5)

-------------------

4(3x-2) - 3(4x-5)

15x - 10 - 8x + 10

------------------

12x - 8 - 12x + 15

7x

---

7

=x