A function f is defined by
f:x→(x+1)/ (x-1),
x is not equal to 1.
If f^2(2)=3f^-1(k),find the value of k.
To find the value of k, let's start by calculating f^2(2) and f^-1(k) based on the given function.
The notation f^2(2) represents evaluating the function f twice for the input value of 2, and f^-1(k) represents finding the inverse function of f and evaluating it for the input value k.
Given that f(x) = (x+1)/(x-1), let's calculate f^2(2):
Step 1: Evaluate f(2):
Substitute x = 2 into the function: f(2) = (2+1)/(2-1) = 3/1 = 3.
Step 2: Evaluate f(f(2)):
Substitute x = 3 (from the previous step) into the function: f(f(2)) = f(3) = (3+1)/(3-1) = 4/2 = 2.
So, f^2(2) = 2.
Now, let's calculate f^-1(k):
Step 1: Swap x and f(x):
Instead of f(x), let's use y, so the equation becomes x = f^(-1)(y).
Step 2: Solve for y:
Swap x and y: x = (y+1)/(y-1).
Cross-multiply: x(y-1) = y+1.
Distribute x: xy - x = y + 1.
Collect like terms: xy - y = x + 1.
Factor out y: y(x-1) = x + 1.
Divide both sides by (x-1): y = (x+1)/(x-1).
So, f^-1(k) = (k+1)/(k-1).
Now, since f^2(2) = 3f^-1(k), we can substitute the values we calculated:
2 = 3 * [(k+1)/(k-1)].
To remove the fraction, let's multiply both sides by (k-1):
2(k-1) = 3(k+1).
Distribute and simplify:
2k - 2 = 3k + 3.
Move all the terms with k to one side:
2k - 3k = 3 + 2.
-k = 5.
Finally, divide by -1 to solve for k:
k = -5.
Therefore, the value of k that satisfies the equation f^2(2) = 3f^-1(k) is k = -5.