Finding Inverse Function by Switching Variables
The question asks for the inverse of the function f(x) = (2x+3)/(x+2), evaluated at x = -5. To find the inverse function, f^(-1)(x), you need to switch the roles of x and y in the equation and then solve for y.
Here's how:
1. Replace f(x) with y:
y = (2x + 3)/(x + 2)
2. Switch x and y:
x = (2y + 3)/(y + 2)
3. Solve for y:
x(y + 2) = 2y + 3
xy + 2x = 2y + 3
xy - 2y = 3 - 2x
y(x - 2) = 3 - 2x
y = (3 - 2x) / (x - 2)
Now that we have the inverse function, we can plug in x = -5 to find the value of f^(-1)(-5):
f^(-1)(-5) = (3 - 2(-5)) / ((-5) - 2)
f^(-1)(-5) = (3 + 10) / (-5 - 2)
f^(-1)(-5) = 13 / (-7)
f^(-1)(-5) = -13/7
So, the value of the inverse function f^(-1) at x = -5 is -13/7.
