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Let's analyze and solve the problems step by step from the images.

### Part (a) - Is \( a > 1 \) or \( 0 < a < 1 \)?

From the graph provided, the exponential curve is decreasing. For an exponential function \( f(x) = a^x \), if \( a > 1 \), the function is increasing. If \( 0 < a < 1 \), the function is decreasing. Since the graph shows a decreasing function, we can conclude that:
- \( 0 < a < 1 \).

### Part (b) - Domain and range of \( f(x) = a^x \)

For any exponential function \( f(x) = a^x \), regardless of the value of \( a \):
- The **domain** is all real numbers: \( (-\infty, \infty) \).
- The **range** is \( (0, \infty) \), because exponential functions never touch or go below the x-axis.

So, the answers are:
- **Domain**: \( (-\infty, \infty) \)
- **Range**: \( (0, \infty) \)

### Part (c) - Sketch the graph of \( g(x) = a^x + 2 \)

This graph will be the same as the original exponential graph \( f(x) = a^x \), but shifted **upward by 2 units**. The entire graph will move up, so the horizontal asymptote will now be at \( y = 2 \) instead of \( y = 0 \).

From the given options, the correct graph would be **option B**, which shows an exponential curve shifted upward.

### Part (d) - Domain and range of \( g(x) = a^x + 2 \)

The domain of \( g(x) = a^x + 2 \) remains the same as \( f(x) = a^x \):
- **Domain**: \( (-\infty, \infty) \).

However, the range changes because the graph is shifted upwards by 2 units:
- The new **range** is \( (2, \infty) \), since the curve never goes below \( y = 2 \).

### Part (e) - Sketch the graph of \( h(x) = a^{x + 3} \)

The function \( h(x) = a^{x+3} \) represents a **horizontal shift** of the graph of \( f(x) = a^x \) **3 units to the left**. 

From the given options, the correct graph would be **option C**, which shows a horizontal shift of the curve to the left.

### Part (f) - Domain and range of \( h(x) = a^{x + 3} \)

The domain of \( h(x) = a^{x+3} \) remains unchanged:
- **Domain**: \( (-\infty, \infty) \).

The range also remains the same as the original exponential function, since a horizontal shift does not affect the range:
- **Range**: \( (0, \infty) \).

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Do you want further explanations on any part? Here are 5 related questions to further expand on this topic:

1. How do vertical shifts affect the range of an exponential function?
2. What happens to the graph of \( f(x) = a^x \) when \( a = 1 \)?
3. How would a horizontal shift of \( f(x) = a^x \) to the right by 5 units be written algebraically?
4. What is the impact on the graph of \( f(x) = a^x \) when you reflect it over the y-axis?
5. Can the range of an exponential function ever be negative?

**Tip**: To sketch transformations, always think about how horizontal and vertical shifts affect the asymptotes and key points on the original graph.

Analyze and solve problems related to exponential functions, including their transformations, domains, and ranges.

Learn how to solve problems involving exponential functions with transformations. Explore horizontal and vertical shifts, calculate domains and ranges, and correctly identify graphs of transformed exponential functions. Suitable for high school students.

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