Question

Give the intervals on which the given function is continuous. $$ f(x)=\sin \left(e^{x}+x^{2}\right) $$

Short answer

The function is continuous everywhere on \( \mathbb{R} \).

Step-by-step solution

Check individual components for continuity

The function given is \( f(x) = \sin(e^x + x^2) \). First, we need to identify the individual components and check if they are continuous. The components are the exponential function \( e^x \), the polynomial \( x^2 \), and the sine function \( \sin(x) \).

Examine the exponential function

The exponential function \( e^x \) is continuous for all real numbers \( x \). This means that for any input \( x \), the output of \( e^x \) is well-defined and continuous.

Examine the polynomial function

The polynomial function \( x^2 \) is a simple monomial, which is continuous for all real numbers \( x \). Polynomials are continuous everywhere.

Examine the sine function

The sine function \( \sin(x) \) is continuous for all real numbers \( x \). This function does not have any jumps, holes, or vertical asymptotes.

Apply the addition rule and composite functions rule

The rules for continuous functions allow that the sum of two continuous functions \( e^x + x^2 \) is also continuous. Since \( e^x + x^2 \) is continuous and composed into a sine function already proven to be continuous, the composition \( \sin(e^x + x^2) \) is continuous wherever \( e^x + x^2 \) is continuous, which is everywhere.

Key concepts

Exponential Functions

Exponential functions like \( e^x \) are essential in mathematics because they model continuous growth or decay processes. Perhaps their most remarkable attribute is their continuity across the entire set of real numbers. An exponential function does not produce jumps or breaks in its graph, which means no matter the input, the output is always consistent and smooth. This property ensures that these functions are predictable and reliable, which is why they are used extensively in fields like finance, biology, and physics.

• Nature of Exponential Growth: Exponential growth leads to an increase that becomes more rapid over time proportional to its current value.
• Natural Base (e): The number \( e \approx 2.718 \) is a mathematical constant that forms the basis of natural logarithms.
• Continuity: Because \( e^x \) is continuous everywhere, any function involving \( e^x \) will also inherit this property through operations like addition or multiplication with other continuous functions.

Polynomial Functions

Polynomial functions are among the simplest and most familiar types of functions. Their defining characteristic is being expressions involving sum of powers of variables multiplied by coefficients, such as \( x^2 \).These functions are continuous for all real numbers. This means that polynomials don’t have sudden jumps, asymptotes, or holes in their graphs. The graph of a polynomial function is a smooth curve without interruptions.

• Basic Structure: Polynomial functions take the form \( a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \).
• Smoothness: The curve generated by polynomial functions is always smooth and unbroken.
• Continuity Everywhere: Because of their structure, polynomial functions are technically continuous across their entire domain of real numbers.

Sine Function

The sine function \( \sin(x) \) is a trigonometric function with a periodic nature—its output values repeat in a predictable cycle. This cycle or wave-like pattern is smooth and never interrupted, meaning the sine function is continuous everywhere over real numbers.This continuity is crucial, particularly in fields like physics and engineering, where sinusoidal functions model waves and oscillations.

• Periodicity: The sine function has a period of \( 2\pi \), meaning it repeats every \( 2\pi \) radians.
• All Real Numbers: No matter what input value you choose, the sine function produces a smooth wave without breaks or gaps.
• Applications: Its continuity makes it ideal for modeling continuous rolls, waves, or oscillations.

Composite Functions

Understanding composite functions is vital when dealing with more complex expressions like \( f(x) = \sin(e^x + x^2) \). A composite function is created when one function is substituted within another. In our example, the function \( e^x + x^2 \) is placed inside the sine function.The rules of continuity allow composite functions to be continuous if each component function is continuous, just like in our earlier explanation: both \( e^x \) and \( x^2 \) are continuous, their sum \( e^x + x^2 \) is continuous, and the sine function is continuous. Thus, \( \sin(e^x + x^2) \) is continuous across all real numbers.

• Continuity Rule: If \( g(x) \) and \( h(x) \) are continuous, then \( g(h(x)) \) is continuous.
• Domain Consistency: Continuity is maintained if the output of the inner function exists within the domain of the outer function.
• Applications: Frequently used in mathematics to build complex models from simpler continuous components.