**Inverse Z-Transform of f(z) = 1/z-1 for k > 0**

To find the inverse Z-Transform of f(z) = 1/(z-1), we can use the properties of the Z-Transform and the known Z-Transform pairs.

**1. Decompose f(z) into partial fractions:**

To start, we can rewrite f(z) as:

f(z) = 1/(z-1) = A/(z-1)

To decompose the fraction, we need to find the value of A. Multiply both sides of the equation by (z-1):

1 = A

Therefore, A = 1. So, we can rewrite f(z) as:

f(z) = 1/(z-1) = 1/(z-1)

**2. Determine the Z-Transform pair for 1/(z-a):**

The Z-Transform pair for the function 1/(z-a) is given by:

Z{1/(z-a)} = a^k u(k)

Where u(k) is the unit step function and a is a constant.

**3. Apply the Z-Transform pair to f(z):**

Using the Z-Transform pair for 1/(z-a), we can rewrite f(z) as:

f(z) = 1/(z-1) = 1^k u(k)

Therefore, the inverse Z-Transform of f(z) is:

z-transform^-1{f(z)} = Z^-1{1/(z-1)} = 1^k u(k)

**4. Simplify the expression using the properties of the unit step function:**

The unit step function is given by:

u(k) = 1 for k >= 0
u(k) = 0 for k < />

Since the condition for k > 0 is specified in the given function, the unit step function becomes:

u(k) = 1 for k > 0
u(k) = 0 for k <=>

Therefore, the inverse Z-Transform of f(z) simplifies to:

z-transform^-1{f(z)} = Z^-1{1/(z-1)} = 1^k u(k) = u(k)

**Final Answer:**

The inverse Z-Transform of f(z) = 1/(z-1) for k > 0 is given by:

z-transform^-1{f(z)} = Z^-1{1/(z-1)} = u(k)

This means that the function f(t) in the time domain is the unit step function u(t).