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The given problem involves analyzing a piecewise function:

\[
f(x) = 
\begin{cases}
e^{-(x-5)} & \text{if } x < 5 \\
e^{x-5} & \text{if } x \geq 5
\end{cases}
\]

### 1. Domain:
The domain of the function consists of all possible values of \(x\) for which the function is defined. Since both pieces of the function are defined for all real numbers, the domain is:

\[
(-\infty, \infty)
\]

### 2. Range:
To find the range, observe the behavior of each piece:

- For \(x < 5\), \(f(x) = e^{-(x-5)}\). As \(x\) approaches negative infinity, \(e^{-(x-5)}\) becomes very large, approaching infinity. As \(x\) approaches 5 from the left, \(e^{-(x-5)}\) approaches 1.

- For \(x \geq 5\), \(f(x) = e^{x-5}\). At \(x = 5\), \(f(5) = 1\). As \(x\) increases, \(e^{x-5}\) approaches infinity.

Therefore, the range is:

\[
(0, \infty)
\]

### 3. Intercepts:
- **Y-intercept**: To find the y-intercept, set \(x = 0\):
  
  \[
  f(0) = e^{-(0-5)} = e^5
  \]

  So, the y-intercept is \((0, e^5)\).

- **X-intercept**: There is no x-intercept because \(f(x)\) is always positive (since exponentials are never zero).

### Summary:
- **Domain:** \((-\infty, \infty)\)
- **Range:** \((0, \infty)\)
- **Intercepts:** \( (0, e^5) \)

Would you like more details on any part? Here are five related questions to deepen understanding:

1. How would the domain and range change if the function were defined differently for \(x < 5\)?
2. What happens to the intercepts if a constant is added to \(f(x)\)?
3. Can you explain why exponentials are never zero?
4. How does the behavior of \(e^{-(x-5)}\) compare with \(e^{x-5}\)?
5. How would the graph change if the piecewise conditions were modified?

**Tip:** When analyzing a piecewise function, always examine each section independently to understand the overall behavior of the function.

Graph the function f(x) = e^{-(x-5)} if x < 5 and e^{x-5} if x ≥ 5. Based on the graph, state the domain, range, and find any intercepts.

Learn how to graph the piecewise function f(x) = e^{-(x-5)} for x < 5 and f(x) = e^{x-5} for x ≥ 5. This solution covers the domain, range, and intercepts of the function, and explains how to analyze piecewise and exponential functions. Suitable for students in Grades 10-12.

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