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When working with exponential functions like \( f(x) = e^x \), it's essential to understand their domain and range. The **domain** refers to all possible input values (\( x \)) the function can accept. For **exponential functions**, the domain is all real numbers because \( e^x \) is defined for every real \( x \). This means you can plug any real number into the function.
On the other hand, the **range** is about the output values (\( f(x) \)) the function can produce. For \( f(x) = e^x \), the range consists of all positive numbers. As \( x \) gets more negative, \( e^x \) gets closer to 0 but never actually reaches it, and as \( x \) gets larger, \( e^x \) grows without bounds.
To summarize:

• **Domain**: \( (-fty, fty) \)
• **Range**: \( (0, fty) \)

An exponential function is a mathematical way of representing situations where values grow or decay at a constant rate. The general form is \( f(x) = a^x \), where \( a \) is a positive constant.
In the given example, \( f(x) = e^x \), \( e \) is approximately equal to 2.718 and is known as Euler's number. It's a special base because it has unique properties that make it invaluable in various mathematical contexts, especially in calculus.
The hallmark of exponential functions is their rapid growth or decay:

• For positive \( x \), the function grows very quickly.
• For negative \( x \), the function decays towards 0 but never actually reaches 0.

Exponential functions are widely used in fields like finance, biology, and physics, often modeling growth processes like population increase or radioactive decay.

Plotting points is crucial for graphing functions by hand. For the function \( f(x) = e^x \), you need to calculate and plot a few key points. This helps you understand the behavior of the function.
Here’s how to do it:
1. Choose values for \( x \). Typical choices might be \(-1\), \(0\), and \(1\).
2. Calculate the corresponding \( f(x) \):

• When \( x = 0 \), \( f(0) = e^0 = 1 \)
• When \( x = 1 \), \( f(1) = e^1 \approx 2.718 \)
• When \( x = -1 \), \( f(-1) = e^{-1} \approx 0.367 \)

3. Plot these points on the coordinate plane.
4. Connect the points smoothly, ensuring the curve approaches the x-axis as \( x \) approaches negative infinity but never touches it.
This whole process will give you a good visual understanding of the function's growth and decay behavior.