Find the inverse of f(x)= (x-3)/(x+2) and f(x)= (8x-4)/(2x+6).
Here's how to do the first one.
(1) solve for x in terms of y. x = f(y)
(2) switch variables x and y.
y = (x-3)/(x+2)
xy + 2y = x - 3
x(y-1) = -3 -2y
x = (2y+3)/(1-y)
The inverse function is
f^-1 (x) = (2x+3)/(1-x)
Do the second problem the same way.
56562
To find the inverse of a function, we need to solve for x in terms of y and then interchange x and y. Let's find the inverse of each of the given functions:
1. f(x) = (x - 3)/(x + 2)
Step 1: Replace f(x) with y.
y = (x - 3)/(x + 2)
Step 2: Interchange x and y.
x = (y - 3)/(y + 2)
Step 3: Solve for y.
Cross-multiply to eliminate the fraction:
x(y + 2) = y - 3
Expand the equation:
xy + 2x = y - 3
Move the y terms to one side:
xy - y = -2x - 3
Factor out y:
y(x - 1) = -2x - 3
Divide both sides by (x - 1):
y = (-2x - 3)/(x - 1)
Therefore, the inverse of f(x) = (x - 3)/(x + 2) is f^(-1)(x) = (-2x - 3)/(x - 1).
2. f(x) = (8x - 4)/(2x + 6)
Step 1: Replace f(x) with y.
y = (8x - 4)/(2x + 6)
Step 2: Interchange x and y.
x = (8y - 4)/(2y + 6)
Step 3: Solve for y.
Cross-multiply to eliminate the fraction:
x(2y + 6) = 8y - 4
Expand the equation:
2xy + 6x = 8y - 4
Move the y terms to one side:
2xy - 8y = -6x - 4
Factor out y:
y(2x - 8) = -6x - 4
Divide both sides by (2x - 8):
y = (-6x - 4)/(2x - 8)
Therefore, the inverse of f(x) = (8x - 4)/(2x + 6) is f^(-1)(x) = (-6x - 4)/(2x - 8).
Note: It's always a good idea to check the inverse by composing the two functions. Just substitute the inverse function into the original function and see if it gives you x as the result.