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The given function is \( f(x) = e^x + \ln(x + 3) \), and we need to determine the range of this function for \( c \in [-2.5, 2] \).

### Steps to solve:

1. **Domain of the function**:
   - The function involves \( \ln(x + 3) \), which requires \( x + 3 > 0 \), so \( x > -3 \).
   - Given \( -2.5 \leq c \leq 2 \), \( x + 3 \geq 0.5 \). Hence, the domain for \( c \) is valid within this range.

2. **Behavior of each component**:
   - \( e^x \) is an increasing exponential function, and for \( x \in [-2.5, 2] \), it ranges from \( e^{-2.5} \) to \( e^2 \).
   - \( \ln(x + 3) \) is a logarithmic function that increases as \( x + 3 \) increases. For \( x \in [-2.5, 2] \), \( x + 3 \in [0.5, 5] \), so \( \ln(x + 3) \) ranges from \( \ln(0.5) \) to \( \ln(5) \).

3. **Finding the range**:
   - At \( x = -2.5 \): \( f(-2.5) = e^{-2.5} + \ln(0.5) \).
   - At \( x = 2 \): \( f(2) = e^2 + \ln(5) \).

4. **Approximations**:
   - \( e^{-2.5} \approx 0.082 \), \( \ln(0.5) \approx -0.693 \). So, \( f(-2.5) \approx 0.082 - 0.693 = -0.611 \).
   - \( e^2 \approx 7.389 \), \( \ln(5) \approx 1.609 \). So, \( f(2) \approx 7.389 + 1.609 = 8.998 \).

Thus, the range of \( f(x) \) for \( c \in [-2.5, 2] \) is approximately \( (-0.611, 8.998) \).

The final answer in exact symbolic form is:

\[
f(c) \in (-0.611, 8.998)
\]

Would you like more details or have any questions?

### Related questions:
1. How does the range of a logarithmic function vary with its argument?
2. What is the behavior of the exponential function at negative values of \( x \)?
3. Can we generalize the range of this function for different intervals of \( x \)?
4. What impact does shifting the logarithmic function by 3 have on its range?
5. How would you approach finding the range of a composite function like \( e^x + \ln(x + 3) \)?

### Tip:
When finding the range of functions that combine exponentials and logarithms, always consider how both components behave over the interval you are interested in.

Find the range for the function f(x) = e^x + ln(x + 3) for some c, where -2.5 ≤ c ≤ 2.

Learn how to find the range of the function f(x) = e^x + ln(x + 3) for values of c in the interval [-2.5, 2]. This problem explores the properties of exponential and logarithmic functions and their behavior over specific intervals. Suitable for high school students in Grades 10-12.

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