A function f is defined by f:x→x+1 over x-1,x is not equal to 1.If f^2(2)=3f^-1(k),find the value of k.
define f^2 (x)
is it ( (x+1)/(x-1) )^2 ??
(x+1)/(x-1)
To find the value of k, we need to break down the given equation step by step. Let's start with finding f^2(2).
The given function is f(x) = x + 1 over x - 1, x is not equal to 1. It means for any value of x (except 1), f(x) will be equal to x + 1 divided by x - 1.
To find f^2(2), we need to substitute 2 into the function twice:
f(2) = 2 + 1 / 2 - 1 = 3 / 1 = 3
Now, substitute f(2) back into the function:
f(f(2)) = f(3) = 3 + 1 / 3 - 1 = 4 / 2 = 2
Now, let's look at the other part of the equation, 3f^-1(k). It means we need to find the inverse function of f(x) and substitute k into it.
To find the inverse function of f(x), we need to interchange the roles of x and f(x), and then solve for f(x):
y = x + 1 / x - 1
Interchanging x and y:
x = y + 1 / y - 1
Now solve for y:
xy - x = y + 1
xy - y = x + 1
y(x - 1) = x + 1
y = (x + 1) / (x - 1)
So, the inverse function is f^(-1)(x) = (x + 1) / (x - 1)
Now, let's substitute k into 3f^(-1)(k):
3f^(-1)(k) = 3 * [(k + 1) / (k - 1)]
Now we can equate both parts of the given equation:
2 = 3 * [(k + 1) / (k - 1)]
To isolate the variable k, let's multiply both sides by (k - 1):
2(k - 1) = 3(k + 1)
2k - 2 = 3k + 3
Subtract 2k and subtract 3 from both sides:
-5 = k
Therefore, the value of k is -5.