Problem 19 - Calculus for Biology and Medicine

Question

In Problems \(15-24\), find the values of \(x \in \mathbf{R}\) for which the given functions are continuous. $$ f(x)=e^{-|x|} $$

Step-by-Step Solution

Answer

The function \( f(x) = e^{-|x|} \) is continuous for all \( x \in \mathbf{R} \).

Step 1: Understanding the Function

The function given is \( f(x) = e^{-|x|} \). The notation \(|x|\) represents the absolute value of \(x\). The absolute value function is continuous everywhere, which affects the exponent in our function \(f(x)\).

Step 2: Exponential Function Continuity

The exponential function \(e^u\) is continuous for any real-valued function \(u\). Hence, \(e^{-|x|}\) is continuous wherever \(-|x|\) is continuous.

Step 3: Continuity of Absolute Value

The absolute value function \(|x|\) is continuous for all real numbers \(x\). Therefore, \(-|x|\) is also continuous for all \(x \in \mathbf{R}\).

Step 4: Conclusion on Continuity

Since both the absolute value function and the exponential function are continuous for all real numbers, the composition \(f(x) = e^{-|x|}\) is also continuous for all \(x \in \mathbf{R}\).

Key Concepts

Understanding the Absolute Value Function

The absolute value function, often symbolized as \(|x|\), is a mathematical way to express the non-negative value of any number. Essentially, it measures the "distance" a number is from zero on the number line.

• For any positive number, the absolute value is the number itself, such that \(|5| = 5\).
• For any negative number, the absolute value is its positive counterpart, so \(|-5| = 5\).
• Finally, the absolute value of zero is simply zero, \(|0| = 0\).

This function is crucial because it helps manage signs in mathematical problems, ensuring expressions involving absolute values remain positive. Importantly, the absolute value function is continuous across all real numbers. This means there are no "jumps" or "breaks" in the graph of \(|x|\), making it smooth and uninterrupted from negative infinity to positive infinity. This property will influence any equation where it appears, maintaining continuity in calculations. In our original problem, the expression of \(|x|\) in the exponent signifies that this section of the function is smooth and predictable for all real numbers.

Exploring the Exponential Function

The exponential function is a powerful mathematical concept represented by \(e^x\), where \(e\) is the base of the natural logarithm, roughly equal to 2.71828. This function expresses how quantities grow or decay exponentially and is prevalent in diverse fields, including finance, physics, and biology.

• Exponential functions involve raising a constant base \(e\) to a variable exponent.
• These functions exhibit continuous growth, implying they neither stop nor jump at any point on the graph.
• What's remarkable is their inherent continuity, which means they produce a smooth curve without breaks.

Of particular interest when exploring continuity is not just the \(e^x\) function itself, but the fact that \(e^u\) remains continuous for any real-valued function \(u\). Thus, as long as \(u\) itself is continuous, \(e^u\) will also be continuous over the same domain. In the function given in the exercise, \(e^{-\|x|}\), the continuity of both the absolute value function \(|x|\) and the exponential function plays a significant role in ensuring the overall continuity of the function \(f(x)\). The combination preserves a seamless graph for every real number.

Delving into Continuity on Real Numbers

Continuity on the set of real numbers \(\mathbf{R}\) means that for any function, the graph will be an unbroken line or curve from the smallest to the largest possible values. This property is essential to predictability and dependability in calculations.

• A function is continuous if small changes in the input \(x\) result in small changes in the output \(f(x)\).
• There should be no gaps, jumps, or asymptotes when moving along the graph.
• A continuous function on real numbers can be drawn without lifting the pencil from the paper.

In the context of the problem given, both components of the function \(f(x) = e^{-|x|}\)—the absolute value and the exponential—are naturally continuous for all real numbers. Thus, their composition—a result of plugging one function into another—retains that continuity. Therefore, implementing these two continuous functions ensures that \(f(x) = e^{-|x|}\) remains continuous across all real numbers, and there are no interruptions in its smooth, flowing graph. Understanding this concept is key in analyzing and predicting the behavior of functions across different domains.