f(x) = (9-8x) / (10x-9)
f^-1(x) =
let y = (9-8x) / (10x-9)
for the inverse, interchange the x and y variables
--->
x = (9-8y)/(10y-9)
10xy - 9x = 9 - 8y
10xy - 8y = 9+9x
y(10x - 8) = 9+9x
y = (9+9x)/(10x - 8)
f^-1 (x) = (9x+9)/(10x-8)
10xy-9x = 9-8y
Wouldn't bringing 8y to the other side be
10xy+8y = 9+9x? not 10xy-8y = 9+9x
I found the answer to be y = (9x+9) / (10x+8)
Please check to see if this is actually the correct answer.
Although I am positive you are correct on the 8y part being +8y
Larry , you are right and observant to catch my error.
So just make the necessary changes
here is a neat trick to make an inverse answer is correct (which I did not do)
pick any "nice" x value and find y
e.g. x = 0
y = (9-0)/(0-9) = -1
now put that into the inverse and you should get back what you started with
so the corrected inverse is
(9x+9)/(10+8)
= 18/18 = 1
To find the inverse of a function, we need to swap the roles of x and y and solve for y.
Let's start by replacing f(x) with y:
y = (9 - 8x) / (10x - 9)
Now, let's swap x and y:
x = (9 - 8y) / (10y - 9)
Next, let's solve for y. To do that, let's cross-multiply:
x(10y - 9) = 9 - 8y
10xy - 9x = 9 - 8y
Let's group the terms with y on one side and the remaining terms on the other side:
10xy + 8y = 9x + 9
Factoring out y on the left side:
y(10x + 8) = 9x + 9
Now, divide both sides of the equation by (10x + 8) to isolate y:
y = (9x + 9) / (10x + 8)
The inverse function of f(x) = (9 - 8x) / (10x - 9) is f^-1(x) = (9x + 9) / (10x + 8).