The value \( f(a, b) = e \), not \( (e, e) \), because \( f(x, y) = e^{xy} \) is a **scalar-valued function** of two variables, not a vector-valued function.

### Here's the breakdown:

1. **Function Definition:**
   - The function \( f(x, y) = e^{xy} \) takes in two inputs, \( x \) and \( y \), and produces a single output, which is a scalar (a single number).
   - For example, if \( x = 1 \) and \( y = 1 \), the output is \( f(1, 1) = e^{1 \times 1} = e \).

2. **Evaluating \( f(a, b) \):**
   - When you evaluate \( f(x, y) \) at the point \( (a, b) = (1, 1) \), you calculate \( f(1, 1) \).
   - Since \( f(x, y) = e^{xy} \), you substitute \( x = 1 \) and \( y = 1 \), leading to \( f(1, 1) = e^{1 \times 1} = e \).

3. **Why Not \( (e, e) \):**
   - \( f(x, y) \) is not a vector, so its output is a single value, not a pair of values.
   - If \( f(x, y) \) were a vector-valued function, then it could produce an output like \( (e, e) \), but here it’s simply \( e \).

So, the correct value for \( f(a, b) \) at \( (1, 1) \) is \( e \), not \( (e, e) \).

Would you like more details on this or have any other questions? Here are five related questions that might be interesting:

1. What is the difference between scalar-valued and vector-valued functions?
2. How do partial derivatives change for scalar versus vector-valued functions?
3. Can you give an example of a vector-valued function and its evaluation at a point?
4. How would you linearize a vector-valued function?
5. What role do gradients play in linear approximations for scalar-valued functions?

**Tip:** Always ensure you understand whether the function is scalar- or vector-valued, as this determines whether its output is a single number or a vector.

Explore why f(a, b) = e and not (e, e) in exponential functions. Learn about scalar-valued functions, vector-valued functions, and the specific function f(x, y) = e^{xy}. Suitable for advanced high school students and anyone interested in mathematical function evaluation.

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