Question

Determine the largest interval over which the given function is continuous. $$ f(x)=\sec ^{-1} x, x \geq 0 $$

Short answer

The function is continuous on \([1, \infty)\).

Step-by-step solution

Understanding the Function's Domain

The function to consider is the inverse secant function, \( f(x) = \sec^{-1}(x) \). The inverse secant is defined for values of \( x \) such that \( |x| \ge 1 \) (to ensure the secant, which is the reciprocal of the cosine function, can have a real value). However, it is important to note that \( x \) is specified to be non-negative, i.e., \( x \ge 0 \). So, \( x \) must satisfy \( x \ge 1 \).

Investigating the Range of Definition

Since \( x \ge 0 \) and we need \( x \ge 1 \) for \( \sec^{-1}(x) \) to be defined, the range of definition becomes \( x \ge 1 \). This is because \( \sec^{-1}(x) \) is not defined for \( 0 \le x < 1 \).

Specifying the Interval of Continuity

The function \( \sec^{-1}(x) \) will be continuous in the domain we have determined, which is for all \( x \) such that \( x \ge 1 \). There are no other discontinuities within this range.

Key concepts

Continuity

Continuity of a function is a fundamental concept in calculus, which ensures that a function behaves nicely without any sudden jumps or breaks. For a function to be continuous, it must meet three key conditions:
1. The function must be defined at that point.
2. The limit of the function as it approaches the point must exist.
3. The limit of the function as it approaches the point must be equal to the function value at that point.
In the context of inverse trigonometric functions like the inverse secant, the continuity plays a crucial role. Ensuring continuity means that whenever we plot a graph of the function over its domain, we get an unbroken line or curve. For the function \( f(x) = \sec^{-1}(x) \), we need to ensure continuity over the interval we determined in the problem statement, specifically for all \( x \geq 1 \). This is the largest interval over which the function \( \sec^{-1}(x) \) remains continuous given that it is non-negative. Since there are no sudden breaks or undefined points in this interval, we can say the function is continuous over this range.

Domain of a Function

The domain of a function is the complete set of possible values of the independent variable, usually represented as \( x \), for which the function is defined. Understanding the domain is critical when working with functions, especially with inverse trigonometric functions like \( f(x) = \sec^{-1}(x) \).
In our example, the inverse secant function \( \sec^{-1}(x) \) has specific criteria for its domain. The secant function, being the reciprocal of the cosine function, is only defined for inputs where \( |x| \ge 1 \). This is because the cosine function itself has maximum and minimum values of 1 and -1, respectively, so its reciprocal must take values outside of these. Since the problem specifies \( x \geq 0 \), the domain is further limited to values where \( x \geq 1 \). Thus, for the function \( \sec^{-1}(x) \) in our exercise, the domain is \( x \geq 1 \). This ensures \( \sec^{-1}(x) \) is defined and real over its specified range.

Range of a Function

The range of a function refers to the set of all possible output values that the function can produce after substituting all values from its domain. Like the domain, knowing the range helps in understanding the behavior of a function and is especially important for inverse functions.
For the inverse secant function \( f(x) = \sec^{-1}(x) \), the range is determined by the output values the inverse secant provides. For \( x \geq 1 \), the range of \( \sec^{-1}(x) \) is typically \([0, \frac{\pi}{2}) \cup (\pi, \frac{3\pi}{2}]\). Essentially, this means the function gives angle measures between and including these bounds, which achieve the secant value specified by \( x \). These values reflect the angles where the original secant function has defined values and thus contributes to forming a complete inverse trigonometric function output.
Understanding the range helps in identifying what outputs are logically possible for the inputs considered, ensuring that all function values make sense within the context of trigonometry.