Let f (x, y) = ln(x + y − 1)
(a) Evaluate f (1, 1)
(b) Evaluate f (e, 1)
(c) Find and sketch the domain of f .
(d) Find the range of f .

Let's solve each part of the question step by step.

### (a) Evaluate \( f(1, 1) \)
Given \( f(x, y) = \ln(x + y - 1) \), substitute \( x = 1 \) and \( y = 1 \):

\[
f(1, 1) = \ln(1 + 1 - 1) = \ln(1) = 0
\]

So, \( f(1, 1) = 0 \).

### (b) Evaluate \( f(e, 1) \)
Substitute \( x = e \) and \( y = 1 \):

\[
f(e, 1) = \ln(e + 1 - 1) = \ln(e) = 1
\]

So, \( f(e, 1) = 1 \).

### (c) Find and sketch the domain of \( f(x, y) \)
The function \( f(x, y) = \ln(x + y - 1) \) requires the argument of the natural logarithm to be strictly positive, i.e.,

\[
x + y - 1 > 0
\]
\[
x + y > 1
\]

Thus, the domain is the set of all \( (x, y) \) pairs such that \( x + y > 1 \). Geometrically, this represents the region above the line \( x + y = 1 \) in the xy-plane (excluding the line itself).

### (d) Find the range of \( f(x, y) \)
Since \( f(x, y) = \ln(x + y - 1) \) and the argument of the logarithm \( (x + y - 1) \) can take values from \( 0^+ \) to \( \infty \) (as \( x + y \) approaches 1 from above or increases without bound), the range of \( f(x, y) \) is all real numbers:

\[
\text{Range of } f(x, y) = (-\infty, \infty)
\]

### Summary:
- **(a)** \( f(1, 1) = 0 \)
- **(b)** \( f(e, 1) = 1 \)
- **(c)** Domain: \( x + y > 1 \)
- **(d)** Range: \( (-\infty, \infty) \)

Do you want details or have any questions?

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**Related Questions:**
1. How do you find the domain of a general logarithmic function?
2. Can we sketch the domain of \( f(x, y) \) in a three-dimensional plot?
3. What is the behavior of \( f(x, y) \) near the boundary \( x + y = 1 \)?
4. How does changing the base of the logarithm affect the range of \( f(x, y) \)?
5. What happens if we add a constant to the function \( f(x, y) \)?

**Tip:** When finding the domain of logarithmic functions, always ensure that the argument inside the logarithm is strictly positive.

Let f (x, y) = ln(x + y − 1)
(a) Evaluate f (1, 1)
(b) Evaluate f (e, 1)
(c) Find and sketch the domain of f .
(d) Find the range of f .

Learn how to evaluate the function f(x, y) = ln(x + y - 1), find its domain and range, and understand its behavior. This problem involves key calculus concepts, including logarithmic functions and multivariable domains. Suitable for college-level calculus students.

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