Graphing Arcsin(x) Functions: Domain, Range & Sketching

Graph, Domain, and Range of arcsin(x)

In what follows, \( \arcsin(x) \) is the inverse function of \( f(x) = \sin(x) \) for \( -\dfrac{\pi}{2} \le x \le \dfrac{\pi}{2} \).

Domain: The range of \( f(x) = \sin(x) \) restricted to \( \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right] \), which is given by the interval \( [-1, 1] \).
Range: The domain of the restricted function \( f \), which is given by the interval \( \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right] \).

The graph of both \( f(x) = \sin(x) \) for \( -\dfrac{\pi}{2} \le x \le \dfrac{\pi}{2} \) and its inverse \( \arcsin(x) \) are shown below:

Graph of sin(x) and arcsin(x)

Table of Key Values for arcsin(x)

A table of values of \( \arcsin(x) \) can be constructed using 3 key points:

\( x \)	\(-1\)	\(0\)	\(1\)
\( y = \arcsin(x) \)	\(-\dfrac{\pi}{2}\)	\(0\)	\(\dfrac{\pi}{2}\)

These 3 key points used to sketch \( \arcsin(x) \) are: \[ \left(-1, -\frac{\pi}{2}\right), \; (0, 0), \; \text{and} \; \left(1, \frac{\pi}{2}\right) \]

Examples & Step-by-Step Solutions

Example 1: Horizontal Shift \( y = \arcsin(x - 2) \)

Problem: Find the domain, range, and graph the function \( y = \arcsin(x - 2) \).

Solution:

The graph of \( y = \arcsin(x - 2) \) will be that of \( \arcsin(x) \) shifted \( 2 \) units to the right.

The domain is found by setting the argument inside the inverse sine between \(-1\) and \(1\): \[ -1 \le x - 2 \le 1 \] Solve the double inequality to find the domain: \[ 1 \le x \le 3 \]

Using the 3 key points via table transformation:

\( x - 2 \)	\(-1\)	\(0\)	\(1\)
\( y = \arcsin(x - 2) \)	\(-\dfrac{\pi}{2}\)	\(0\)	\(\dfrac{\pi}{2}\)
\( x \)	\(1\)	\(2\)	\(3\)

The value of \( x \) is calculated from \( x - 2 \) (e.g., when \( x - 2 = -1 \), solving for \( x \) gives \( x = 1 \)).

Domain: \( [1, 3] \)
Range: \( \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right] \)

Graph of y = arcsin(x - 2)

Example 2: Vertical Stretch & Shift \( y = 2\arcsin(x + 1) \)

Problem: Find the domain, range, and graph the function \( y = 2\arcsin(x + 1) \).

Solution:

We evaluate the transformations using the 3 key reference points, computing the intermediate values and final coordinates:

\( x + 1 \)	\(-1\)	\(0\)	\(1\)
\( \arcsin(x + 1) \)	\(-\dfrac{\pi}{2}\)	\(0\)	\(\dfrac{\pi}{2}\)
\( y = 2\arcsin(x + 1) \)	\(-\pi\)	\(0\)	\(\pi\)
\( x \)	\(-2\)	\(-1\)	\(0\)

Domain: \( [-2, 0] \)
Range: \( [-\pi, \pi] \)

Graph of y = 2arcsin(x+1)

Geometric description: The graph is that of \( \arcsin(x) \) shifted one unit to the left and stretched vertically by a factor of \( 2 \).

Example 3: Reflection & Horizontal Shift \( y = -\arcsin(x - 1) \)

Problem: Find the domain, range, and graph the function \( y = -\arcsin(x - 1) \).

Solution:

We track the key points including the vertical reflection:

\( x - 1 \)	\(-1\)	\(0\)	\(1\)
\( \arcsin(x - 1) \)	\(-\dfrac{\pi}{2}\)	\(0\)	\(\dfrac{\pi}{2}\)
\( y = -\arcsin(x - 1) \)	\(\dfrac{\pi}{2}\)	\(0\)	\(-\dfrac{\pi}{2}\)
\( x \)	\(0\)	\(1\)	\(2\)

Domain: \( [0, 2] \)
Range: \( \left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right] \)

Graph of y = -arcsin(x-1)

Geometric description: The graph is that of \( \arcsin(x) \) shifted one unit to the right and reflected across the \( x \)-axis.

Graphing Arcsin(x) Functions

Graph, Domain, and Range of arcsin(x)

Table of Key Values for arcsin(x)

Examples & Step-by-Step Solutions

More References and Links to Graphing