Problem 51 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 49-54, determine the largest interval over which the given function is continuous. $$ f(x)=\sin ^{-1} x $$

Step-by-Step Solution

Answer

The function \( \sin^{-1} x \) is continuous over \([-1, 1]\).

Step 1: Identify the Function

The function given is the inverse sine function, denoted as \( f(x) = \sin^{-1} x \). This function is known to have a specific domain over which it is defined and continuous.

Step 2: Recall the Domain of \( \sin^{-1} x \)

The inverse sine function \( \sin^{-1} x \) is defined for values of \( x \) that lie in the interval \([-1, 1]\). This is because the sine of any angle in the real numbers only produces values from \(-1\) to \(1\).

Step 3: Determine Continuity Within Domain

Within its domain, \([-1, 1]\), the function \( f(x) = \sin^{-1} x \) is continuous. This means it does not have any breaks, jumps, or asymptotes within this interval.

Key Concepts

Continuity of Functions

Continuity is a fundamental property of functions which means that the function has no interruptions or sudden jumps. In simple terms, a function is continuous when you can draw its graph without lifting your pen off the paper. For a function to be continuous at a point, the following three conditions must be met:

• The function is defined at that point.
• The limit of the function exists at that point.
• The value of the function at that point equals the limit of the function as it approaches that point.

For the inverse sine function, denoted as \( \sin^{-1} x \), continuity is found over its entire domain, which is the interval \([-1, 1]\). This means within this interval, you can expect the function to behave predictably, with no sudden jumps or breaks.

Function Domain

The domain of a function refers to the set of all possible input values (usually \( x \)) for which the function is defined. An intuitive way to think about it is the collection of 'allowable' values for \( x \). For the inverse sine function \( \sin^{-1} x \), the domain is restricted to the interval \([-1, 1]\). This restriction comes from the nature of the sine function. Since the sine function produces values between \(-1\) and \(1\), the inverse sine, or arc sine, can only take those values as inputs. Therefore, when working with \( \sin^{-1} x \), it's important to remember that inputting values outside of the domain \([-1, 1]\) will lead you to undefined outputs. Always check the domain before solving problems involving inverse trigonometric functions.

Trigonometric Functions

Trigonometric functions are essential in mathematics and are based on the relationships of angles with their corresponding sides in a right triangle. The primary trigonometric functions are sine, cosine, and tangent. Each of these functions has an inverse: sine inverse \( \sin^{-1} \), cosine inverse \( \cos^{-1} \), and tangent inverse \( \tan^{-1} \).

• Inverse Sine: \( \sin^{-1} x \) is the angle whose sine is \( x \).
• Inverse Cosine: \( \cos^{-1} x \) is the angle whose cosine is \( x \).
• Inverse Tangent: \( \tan^{-1} x \) is the angle whose tangent is \( x \).

The inverse sine function, \( \sin^{-1} x \), specifically returns an angle in the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \), and it is defined for \( x \) values between \(-1\) and \(1\). Both trigonometric functions and their inverses are used extensively in calculus and beyond. They are crucial in solving equations, analyzing periodic phenomena, and understanding oscillations.