Problem 45 - Single Variable Calculus

Question

Find the interval(s) where \(f\) is continuous. \(f(x)=\sin e^{\sqrt{x}}\)

Step-by-Step Solution

Answer

The function \(f(x) = \sin e^{\sqrt{x}}\) is continuous in the interval \([0, +\infty)\).

Step 1: Identify the domain of the inner function (square root)

The square root function is continuous for non-negative real numbers. Thus, we can identify its domain as \(x \geq 0\).

Step 2: Identify the domain of the composition of inner function and exponentiation function

Since exponentiation is continuous for any real number, we do not need to restrict the domain of \(e^{\sqrt{x}}\) beyond what we have determined in step 1. Therefore, the domain of \(e^{\sqrt{x}}\) is \(x \geq 0\).

Step 3: Identify the domain of the composition of all functions

Since the sine function is continuous for any real number as input, there is no need to restrict the domain of \(f(x) = \sin e^{\sqrt{x}}\) beyond what we have determined in step 2. Thus, the domain of \(f(x)\) is \(x \geq 0\).

Step 4: Identify the interval where the function is continuous

Since we have determined the domain of \(f(x)\) to be \(x \geq 0\) and all the composing functions (square root, exponentiation, and sine) are continuous within their respective domains, we can say that the function \(f(x) = \sin e^{\sqrt{x}}\) is continuous for its entire domain of \(x \geq 0\). So, the interval where the function \(f(x)\) is continuous is \([0, +\infty)\).

Key Concepts

Domain of Functions

When discussing the domain of functions, we are essentially talking about the set of all possible inputs for which the function gives a valid output. For a given function, examining which values can be "plugged in" without causing mathematical issues such as division by zero or the square root of a negative, determines its domain.
For instance in the case of the function \(f(x) = \sin e^{\sqrt{x}}\), let's break down its domain:

• Square Root Function: The term \(\sqrt{x}\) is defined only for non-negative numbers (i.e., \(x \geq 0\)), since the square root of a negative number is not real.
• Exponentiation Function: The \(e^{\sqrt{x}}\) function is defined and continuous for all real numbers; thus, it operates seamlessly on the outputs of the square root function.
• Sine Function: \(\sin(y)\) is defined for all real numbers \(y\), offering no further restrictions on the composition of functions.

Therefore, since each segment of \(f(x)\) seamlessly accepts non-negative numbers, the overall domain is \(x \geq 0\). This means any real number equal to or greater than zero can be an input for the function.

Composition of Functions

Composing functions means placing one function inside another. This process helps create more complex expressions from simpler ones. The composition of functions is denoted by \((f \circ g)(x) = f(g(x))\), meaning you apply the function \(g\), and then \(f\) on the result of \(g\).
In our example, \(f(x) = \sin e^{\sqrt{x}}\) involves multiple nested functions:

• Inner Function: \(\sqrt{x}\) - calculates the square root.
• Middle Function: \(e^{\sqrt{x}}\) - takes the result of \(\sqrt{x}\) and uses it as the exponent for \(e\).
• Outer Function: \(\sin(y)\) - takes \(e^{\sqrt{x}}\) and computes the sine of that value.

By understanding each function's role and domain, you can ensure the overall composition is well-defined. Each step's output acts as a valid input for the next, providing a smoothly defined transition ensuring the whole composite function remains valid for \(x \geq 0\). So, by crafting complex functions through composition, we maintain the integrity and limit the domain to acceptable values.

Continuous Functions

A function is continuous when you can draw its graph without lifting your pencil from the paper. In mathematical terms, this means at every point in the domain, the function's limit equals the function value. Continuous functions do not have jumps, breaks, or holes.
In the context of \(f(x)=\sin e^{\sqrt{x}}\), we consider:

• Square Root Function: \(\sqrt{x}\) is continuous for \(x \geq 0\) because it steadily increases without interruption in its domain.
• Exponentiation Function: \(e^{y}\) is known for its continuous nature over all real numbers.
• Sine Function: \(\sin(y)\) is periodic and continuous over the entire number line.

Because each function involved is continuous within its domain, the composition \(f(x)=\sin e^{\sqrt{x}}\) remains continuous for all \(x \geq 0\). Thus, the function does not stumble in any area, maintaining a smooth, unbroken path throughout the interval \([0, +\infty)\). Thus, within this interval, the continuity ensures that consistent behavior is maintained without surprises.