find the inverse of the function, graph the inverse function, determine the domain, range and asympotoes

f(x)= e^-x/(2)+4

To find the inverse of a function, we swap the roles of x and y and solve for y. Let's start with the function f(x) = e^(-x/2) + 4.

Step 1: Swap x and y:
x = e^(-y/2) + 4

Step 2: Solve for y:
x - 4 = e^(-y/2)

Step 3: Take the natural logarithm of both sides:
ln(x - 4) = -y/2

Step 4: Multiply by -2 to isolate y:
-2ln(x - 4) = y

So the inverse function of f(x) is:
f^(-1)(x) = -2ln(x - 4)

Now, let's graph the inverse function f^(-1)(x).

To determine the domain of the inverse function, we need to consider the range of the original function. In this case, since we are using the natural logarithm, the expression inside the logarithm (x - 4) must be greater than zero. So the domain of f^(-1)(x) is x > 4.

To determine the range of the inverse function, we need to consider the domain of the original function. Since the original function is the exponential function e^(-x/2), its range is all positive real numbers. Therefore, the range of the inverse function f^(-1)(x) is y > 0.

As for asymptotes, since the original function doesn't have any vertical asymptotes, the inverse function also doesn't have any vertical asymptotes. However, the original function approaches 0 as x approaches positive infinity, and as a result, the inverse function approaches negative infinity as x approaches positive infinity. So the inverse function has a horizontal asymptote at y = -∞.

To summarize:
- The inverse function of f(x) = e^(-x/2) + 4 is f^(-1)(x) = -2ln(x - 4).
- The domain of f^(-1)(x) is x > 4.
- The range of f^(-1)(x) is y > 0.
- The inverse function has a horizontal asymptote at y = -∞.