Problem 54 - Calculus Early Transcendentals: Pearson New International Edition

Question

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

Step-by-Step Solution

Answer

The largest interval of continuity for \( f(x) = \operatorname{sech}^{-1}(x) \) is \( (0, 1] \).

Step 1: Understand the Function

The function given is the inverse hyperbolic secant, denoted as \( f(x) = \operatorname{sech}^{-1}(x) \). To determine the continuity, we need to understand the domain of this function.

Step 2: Identify the Domain of Inverse Hyperbolic Secant

The inverse hyperbolic secant function \( \operatorname{sech}^{-1}(x) \) is defined for \( 0 < x \leq 1 \). This is because the hyperbolic secant function \( \operatorname{sech}(x) = \frac{2}{e^x + e^{-x}} \) takes values in the interval \((0, 1]\) for all real \( x \).

Step 3: Determine the Continuity

Since \( \operatorname{sech}^{-1}(x) \) is defined and differentiable over the interval \( (0, 1] \), it is continuous over this interval. The function cannot be continuous at \( x = 0 \) because it is not defined there.

Step 4: Determine the Largest Interval of Continuity

Based on the domain, the largest interval over which the function \( f(x) = \operatorname{sech}^{-1}(x) \) is continuous is \( (0, 1] \).

Key Concepts

Inverse Hyperbolic Functions

Inverse hyperbolic functions are mathematical functions that serve as the inverse of the hyperbolic functions like sinh, cosh, and tanh. Just like their trigonometric counterparts, these functions are important in calculus and other areas of mathematics.
The inverse hyperbolic secant, denoted as \( \operatorname{sech}^{-1}(x) \), is the focus here. It reverses the hyperbolic secant function \( \operatorname{sech}(x) = \frac{2}{e^x + e^{-x}} \). The domain for \( \operatorname{sech}(x) \) spans \( (0, 1] \), covering all the resultant values that can be sent back into \( \operatorname{sech}^{-1} \).
Understanding these functions is key for solving calculus problems that involve hyperbolic characteristics. They are useful as they help model certain physical scenarios in engineering and physics, similar to how exponential and trigonometric functions are used.

Continuity

Continuity is a fundamental concept in calculus that deals with the smoothness of a function's graph. A function is said to be continuous on an interval if, moving within that interval, there are no breaks, holes, or jumps in the graph.
For \( \operatorname{sech}^{-1}(x) \) to be continuous over an interval, the function must be defined and differentiable all along that interval. In the problem given, it was determined that the function \( \operatorname{sech}^{-1}(x) \) is continuous on the interval \( (0, 1] \) because it is both defined and without breaks in this range.
Continuous functions are nice to work with because of their predictable behavior. They ensure there are no sudden changes in the value of the function, which is crucial for problems in physics and engineering where precision and predictability are paramount.

Function Domains

The domain of a function is the set of all possible inputs (x-values) for which the function is defined. Determining the domain is a prerequisite to understanding where the function behaves normally without any undefined points or anomalies.
For the inverse hyperbolic secant function \( \operatorname{sech}^{-1}(x) \), the domain is explicitly set as \( 0 < x \leq 1 \). This limitation stems from the inverse relationship it holds with \( \operatorname{sech}(x) \), which naturally produces results only within this range from inputs that cover all real numbers.
Understanding the domain is crucial because it tells us the scope for x-values that we can safely use in any calculations or graphs. Any use of \( x \) outside this specified range might lead to undefined results or mathematical errors, making it vital to respect these boundaries to maintain mathematical integrity.