Problem 22 - Calculus for Biology and Medicine

Question

In Problems 15-24, find the values of \(x \in\) R for which the given functions are both defined and continuous. $$ f(x)=\exp [\sqrt{x-1}] $$

Step-by-Step Solution

Answer

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

Step 1: Understand Function Definition

The function given is \(f(x) = \exp [\sqrt{x-1}]\). This function is composed of an exponential function and a square root function. Both must be defined for \(f(x)\) to exist.

Step 2: Determine Domain of Inner Function

The inner function \(\sqrt{x-1}\) requires that \(x-1 \geq 0\) for the square root to be defined over the real numbers. Thus, \(x \geq 1\).

Step 3: Ensure Continuity

The exponential function \(\exp(u)\) is continuous for all \(u\) in the real numbers. Since \(\sqrt{x-1}\) is continuous for \(x \geq 1\) as a function composed of a continuous square root transformation, \(f(x)\) is continuous for \(x \geq 1\) as well.

Key Concepts

Exponential Function

An exponential function is one of the most important functions in mathematics, particularly in the study of growth and decay processes. It is generally written as \( f(x) = a^x \), where \( a \) is a constant and it serves as the base of the function. The most commonly used base is Euler's number \( e \), approximately equal to 2.718. This special exponential function is written as \( \exp(x) = e^x \).
The exponential function has several remarkable properties:

• It is continuous and differentiable over all real numbers.
• The derivative of \( \exp(x) \) is itself \( \exp(x) \), which is quite unique.
• It rapidly increases as \( x \) becomes larger and exponentially decreases as \( x \) becomes negative.

In equation \( f(x) = \exp[\sqrt{x-1}] \), the outer part \( \exp(u) \) ensures that for the values of \( u \), which are real numbers, the exponential function remains continuous.

Square Root Function

The square root function is another fundamental function in mathematics. It is represented as \( g(x) = \sqrt{x} \). This function plays a crucial role in algebra and calculus, especially when dealing with quadratic equations and their solutions.
The square root function has specific properties that dictate its behavior:

• It is only defined for non-negative real numbers. This is because the square root of a negative number is not a real number under standard real analysis.
• The function is continuous and increases slowly as \( x \) increases.

In the given function \( f(x) = \exp[\sqrt{x-1}] \), \( \sqrt{x-1} \) acts as an inner function, requiring that \( x-1 \geq 0 \). This logically leads to the domain constraint that \( x \geq 1 \) for the square root to be defined and continuous within real numbers.

Domain of a Function

The domain of a function is essential in mathematics, as it specifies all possible input values (or \( x \)-values) that will yield a valid output. In simple terms, it is the collection of all values for which the function is defined and can produce corresponding outcomes.
To determine the domain:

• Consider any restrictions based on mathematical operations involved, like dividing by zero or taking the square root of a negative number.
• Identify intervals where composite functions, like in \( f(x) = \exp[\sqrt{x-1}] \), remain both defined and continuous.

The function in the exercise has a domain where both the square root and exponential sections operate within real numbers. Thus, the domain is derived from ensuring \( \sqrt{x-1} \) is valid, yielding \( x \geq 1 \). This domain ensures that the overall function \( f(x) \) remains defined and continuous across this interval.