Find the inverse of the function

f(x) = 5 – e^(x-1).

Determine the domain, range, and asymptotes of the function and its inverse function. Please show all of your work.

To find the inverse of the function f(x) = 5 – e^(x-1), let's follow these steps:

1. Replace f(x) with y: y = 5 – e^(x-1).
2. Swap x and y: x = 5 – e^(y-1).
3. Solve for y: x - 5 = -e^(y-1).
4. Multiply both sides by -1: 5 - x = e^(y-1).
5. Take the natural logarithm of both sides to eliminate the exponential: ln(5 - x) = y - 1.
6. Add 1 to both sides to isolate y: ln(5 - x) + 1 = y.

Therefore, the inverse of the function f(x) is:

f^(-1)(x) = ln(5 - x) + 1.

Now let's determine the domain, range, and asymptotes of the function and its inverse.

Domain of f(x): Since f(x) includes an exponential term e^(x-1), the domain is all real numbers.

Range of f(x): The range of f(x) depends on the exponential term e^(x-1). Since e^x is always positive, and f(x) subtracts this positive term from 5, the range of f(x) is (-∞, 5].

Asymptotes of f(x): Since f(x) includes an exponential term e^(x-1), there are no asymptotes.

Domain of f^(-1)(x): The domain of f^(-1)(x) is all real numbers, as the logarithm function ln(5 - x) is defined for all x ≤ 5.

Range of f^(-1)(x): The range of f^(-1)(x) is also all real numbers, as the logarithm function ln(5 - x) allows x to approach 5 but never equals it.

Asymptotes of f^(-1)(x): Since f^(-1)(x) includes the logarithm function ln(5 - x), there is a vertical asymptote at x = 5.

To find the inverse of a function, let's denote the inverse function as f^(-1)(x) and solve for x in terms of f.

Step 1: Replace f(x) with y.
y = 5 - e^(x-1).

Step 2: Interchange x and y.
x = 5 - e^(y-1).

Step 3: Solve for y.
x - 5 = - e^(y-1).

Step 4: Multiply both sides by -1 to isolate the exponential term.
5 - x = e^(y-1).

Step 5: Take the natural logarithm of both sides.
ln(5 - x) = y - 1.

Step 6: Solve for y.
y = ln(5 - x) + 1.

Therefore, the inverse of the function f(x) = 5 - e^(x-1) is f^(-1)(x) = ln(5 - x) + 1.

Next, let's determine the domain, range, and asymptotes of the function f(x) and its inverse function f^(-1)(x).

Domain of f(x):
The function f(x) contains an exponential term e^(x-1). Exponential functions are defined for all real values of x. However, since there is a -1 in the exponent, x cannot be less than 1. Therefore, the domain of f(x) is x >= 1.

Range of f(x):
The exponential term e^(x-1) is always positive. Since 5 is subtracted from this positive term, the range of f(x) is all x such that f(x) <= 5.

Asymptote of f(x):
To find the asymptote of f(x), let's find the limit as x approaches positive or negative infinity.
lim (x->∞) f(x) = lim (x->∞) (5 - e^(x-1)).
Since the exponential term e^(x-1) grows much faster than 5 as x approaches infinity, the limit is negative infinity. Therefore, the asymptote of f(x) as x approaches infinity is y = -∞.

Similarly, when x approaches negative infinity, the exponential term approaches zero, and f(x) approaches 5. Therefore, the asymptote of f(x) as x approaches negative infinity is y = 5.

For the inverse function f^(-1)(x), the domain, range, and asymptotes are inverted compared to f(x):

Domain of f^(-1)(x):
Since f^(-1)(x) is the inverse of f(x), its domain is the range of f(x). Therefore, the domain of f^(-1)(x) is all x such that f^(-1)(x) <= 5.

Range of f^(-1)(x):
Similarly, the range of f^(-1)(x) is the domain of f(x). Therefore, the range of f^(-1)(x) is x >= 1.

Asymptotes of f^(-1)(x):
The asymptotes of f^(-1)(x) are the same as f(x) but inverted. Therefore, as x approaches positive infinity, f^(-1)(x) approaches y = 5, and as x approaches negative infinity, f^(-1)(x) approaches y = -∞.

In summary:
- The inverse of the function f(x) = 5 - e^(x-1) is f^(-1)(x) = ln(5 - x) + 1.
- The domain of f(x) is x >= 1, and its range is f(x) <= 5.
- The asymptote of f(x) as x approaches negative infinity is y = 5, and as x approaches positive infinity, y = -∞.
- The domain of f^(-1)(x) is f^(-1)(x) <= 5, and its range is x >= 1.
- The asymptote of f^(-1)(x) as x approaches -∞ is y = -∞, and as x approaches +∞, y = 5.