The function f is defined as f(x)=(2x+1)/(x-3). Find the value of k so that the inverse of f is f^-1(x)=(3x+1)/(x-k).
We'll determine the inverse function f^-1(x).
Let f(x)=y, such as:
y = (2x+1)/(x-3)
We'll multiply both sides by (x-3):
xy - 3y = 2x+1
We'll move 3y to the right:
xy = 2x + 3y +1
We'll subtract 2x both sides:
xy - 2x = 3y + 1
We'll factorize by x to the left:
x(y - 2) = 3y + 1
We'll divide by (y-2):
x = (3y+1)/(y-2)
The inverse function is: f^-1(x) = (3x+1)/(x-2)
Comparing with the given expression of f^-1(x) = (3x+1)/(x-k), we'll identify k = 2.