Please find the inverse function of f.
Let f(x)=(x+3)/(2x-5)
What is the domain of f? What is the domain of the inverse function?
If 2x-5 = 0, we have a problem
then x = 5/2
so domain is all x except x = 5/2
(2x-5)y = (x+3)
2 x y - 5 y = x + 3
x(2y-1) = 5y+3
f(y) = (5y+3)/(2y-1)
This one blows up when y = 1/2
Let y = f(x)
y = (x + 3)/(2x - 5)
Switch x and y.
x = (y + 3)/(2y - 5)
Multiply both sides by (2y - 5).
2xy - 5x = y + 3
Subtract 3 from both sides.
2xy - 5x - 3 = y
Subtract 2xy from both sides.
-5x - 3 = -2xy + y
Factor the right side.
-5x - 3 = y(-2x + 1)
Divide both sides by (-2x + 1).
(-5x - 3)/(-2x + 1) = y
(5x - 3)/(2x + 1) = y
Now replace y with the inverse function notation: f^(-1) x.
f^(-1) x = (5x - 3)/(2x + 1)
To find the inverse function of f(x), we can follow these steps:
Step 1: Start with the equation for f(x): f(x) = (x + 3)/(2x - 5).
Step 2: Replace f(x) with y: y = (x + 3)/(2x - 5).
Step 3: Swap x and y to interchange the dependent and independent variables: x = (y + 3)/(2y - 5).
Step 4: Solve the equation for y.
a) Multiply both sides of the equation by 2y - 5 to eliminate the denominator: (2y - 5) * x = y + 3.
b) Distribute x to both terms on the left side: 2xy - 5x = y + 3.
c) Move all terms involving y to one side: 2xy - y = 5x + 3.
d) Factor out y: y(2x - 1) = 5x + 3.
e) Divide both sides by (2x - 1) to solve for y: y = (5x + 3)/(2x - 1).
Step 5: Replace y with f^(-1)(x) to get the inverse function:
f^(-1)(x) = (5x + 3)/(2x - 1).
So, the inverse function of f(x) = (x + 3)/(2x - 5) is f^(-1)(x) = (5x + 3)/(2x - 1).