Problem 102 - Algebra and Trigonometry

Question

Without drawing a graph, describe the behavior of the graph of \(y=\sin ^{-1} x .\) Mention the function's domain and range in your description.

Step-by-Step Solution

Answer

The inverse sine function, \(y=\sin^{-1}x\), is defined for all \(x\) such that \(-1 \leq x \leq 1\) and its range is \(-\pi/2 \leq y \leq \pi/2\) or \(-90^{\circ} \leq y \leq 90^{\circ}\) in degrees. The function is strictly increasing throughout its domain, starting at \(-1, -\pi/2\) and ending at \(1, \pi/2\). The graph has a horizontal tangent at \(x=-1\) and \(x=1\) and a highest slope at \(x=0\). The graph is symmetric with respect to the line \(y=x\).

Step 1: Understanding Inverse Sine function

The function \(y=\sin^{-1}x\) is the inverse sine function, also known as arcsin. It is used to determine the angle that produces a given sine value. Unlike the sine function, which is periodic and oscillatory in nature, the inverse sine function is not.

Step 2: Defining the Domain

The domain of the inverse sine function is the set of all real numbers that sine can take. Since the value of a sine function ranges from -1 to 1, the domain of the inverse sine function is thus, \(-1 \leq x \leq 1\). This means that the function is defined for all \(x\) between -1 and 1, inclusive of -1 and 1.

Step 3: Defining the Range

The range of the inverse sine function is the set of all possible output values, which is the set of all angles that the sine function can take. Since sine is defined for angles between \(-\pi/2\) and \(\pi/2\) (in radians), the range of the inverse sine function is \(-\pi/2 \leq y \leq \pi/2\). In degrees, this is equivalent to \(-90^{\circ} \leq y \leq 90^{\circ}\).

Step 4: Describing its Behavior

The graph of \(y=\sin^{-1}x\) starts at \(-1, -\pi/2\) and ends at \(1, \pi/2\). The function is strictly increasing throughout its domain, with a horizontal tangent (slope of 0) at \(x=-1\) and \(x=1\) (when \(y=-\pi/2\) and \(y=\pi/2\), respectively), and a highest slope (tangent) at \(x=0\). The graph is symmetric with respect to the line \(y=x\).