Sketch the graph for y = sin^-1(sin x)

To sketch the graph of the function y = sin^-1(sin x), we need to understand how the inverse sine function works in conjunction with the sine function.

Understanding the Functions

The sine function, sin x, oscillates between -1 and 1 for all real numbers. The inverse sine function, sin^-1(y), returns values in the range of [-π/2, π/2].

Key Points to Consider

• The output of sin x will be between -1 and 1.
• The sin^-1(y) function will only return values from [-π/2, π/2].
• For values of x outside the range of [-π/2, π/2], the function will reflect the periodic nature of sine.

Graphing the Function

1. **For x in the interval [-π/2, π/2]:** The graph of y = sin^-1(sin x) will be a straight line, as the output equals the input.

2. **For x outside this interval:** The graph will repeat the values of y based on the periodicity of the sine function, effectively "folding" back into the range of [-π/2, π/2].

Visual Representation

When sketching, you will see a wave-like pattern that stays within the bounds of [-π/2, π/2] for all values of x. The graph will look like a series of line segments connecting the peaks and troughs of the sine wave, reflecting the periodic nature of the sine function.

Final Thoughts

By understanding the behavior of both sine and its inverse, you can accurately sketch the graph of y = sin^-1(sin x). Remember to focus on the intervals and how the function behaves outside the principal range.