Find a formula for the inverse of the function
f(x)=1+8x/8−5x.
f^[−1](Q) =
f = (1+8x)/(8-5x)
swap variables and you have
x = (1+8f)/(8-5f)
x(8-5f) = 1+8f
8x - 5xf = 1 + 8f
-5xf - 8f = 1 - 8x
-(5x+8)f = (1-8x)
f = (8x-1)/(5x+8)
To find the inverse of a function, we need to exchange the roles of x and y and solve for y.
Let's start by replacing f(x) with y:
y = 1 + (8x / (8−5x))
Now, let's switch x and y:
x = 1 + (8y / (8−5y))
Next, we'll solve for y.
To start, let's clear the fraction by multiplying both sides by (8−5y):
x(8−5y) = 1 + (8y)
Expanding the left side:
8x − 5xy = 1 + 8y
Rearranging the terms to isolate y on one side:
-5xy - 8y = 1 - 8x
Factoring out y:
y(-5x - 8) = 1 - 8x
Dividing both sides by (-5x - 8):
y = (1 - 8x) / (-5x - 8)
Therefore, the formula for the inverse of the function f(x) = 1 + (8x / (8−5x)) is:
f^[-1](x) = (1 - 8x) / (-5x - 8)