Problem 22 - 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. $$ f(x)=\exp [-\sqrt{x-1}] $$

Step-by-Step Solution

Answer

The function is continuous for \(x \geq 1\).

Step 1: Understand the Domain for Square Root Functions

The square root function, \(\sqrt{x-1}\), is defined only for values where its argument is zero or positive. Therefore, for the expression \(x-1\), we need \(x-1 \geq 0\). This tells us that \(x \geq 1\).

Step 2: Consider the Properties of the Exponential Function

The exponential function, \(\exp(y)\), is continuous for all real numbers \(y\). Therefore, as long as \(y = -\sqrt{x-1}\) is defined, the exponential function will be continuous.

Step 3: Define the Continuity of the Composite Function

Since the square root function is defined for \(x \geq 1\) and the exponential function is continuous across all real numbers, the composite function \(f(x) = \exp(-\sqrt{x-1})\) is continuous for all \(x\) in the domain where it is defined, which is \(x \geq 1\).

Key Concepts

Exponential Function

The exponential function is a fundamental mathematical concept that grows rapidly. When we write \(\exp(y)\), it means the same thing as \(e^y\), where \(e\) is approximately 2.71828. This number is irrational and often called Euler's number. The exponential function is hugely important due to its unique properties:

• Continuity: The exponential function is continuous everywhere on the real number line. This means no sudden jumps or breaks occur, regardless of the value of \(y\).
• Always positive: \(\exp(y)\) is always greater than zero, regardless of the input \(y\).
• Rapid growth: As \(y\) increases, \(\exp(y)\) grows very quickly. Conversely, as \(y\) becomes more negative, \(\exp(y)\) approaches zero but never reaches it.

In the function \(f(x) = \exp(-\sqrt{x-1})\), the exponential function ensures that the outputs remain continuous, as long as \(y = -\sqrt{x-1}\) is defined.

Square Root Function

A square root function, written as \(\sqrt{x}\), is a function that gives us a number that, when squared, equals the original \(x\). It's a fundamental building block of algebra. Here are some key features:

• Non-negative inputs: The square root function is only defined for non-negative numbers. If \(y = \sqrt{x-1}\), then \(x-1\) must be greater than or equal to zero, which means \(x \geq 1\).
• Non-negative outputs: The outputs of \(\sqrt{x}\) are also non-negative, which is why it’s shown as "cpm" when referring to possible positive and negative roots.
• Continuous but not globally defined: While the function is continuous for its defined set, not all real numbers can be plugged into a square root function.

Thus, in the composite function \(f(x) = \exp(-\sqrt{x-1})\), \(\sqrt{x-1}\) restricts the domain to values of \(x\) greater than or equal to 1.

Domain of a Function

Understanding the domain of a function is key to determining where a function is correctly defined. The domain refers to all the possible input values (often

Domain of a Function

represented as \(x\)) that you can put into a function without hitting mathematical complications, like division by zero or taking the square root of a negative number.

• Defining Criteria: When considering the domain of a composite function like \(f(x) = \exp(-\sqrt{x-1})\), you need to consider each part: ensure that the square root is defined, which means \(x \geq 1\).
• Composite Functions: Once you have established where each component of a composite function is defined, plug those values into each part to confirm they all produce valid outputs.
• Implications for Continuity: The domain directly affects continuity; if a value is outside the domain, the function ceases to be continuous at that point.

In summary, for the function \(f(x) = \exp(-\sqrt{x-1})\), we find the domain by looking at the inside of the square root. Because it needs to be zero or positive, we conclude that \(x \geq 1\) is the domain ensuring continuity across those values.