Image of Composite Functions

This lecture explores the image set of a function defined on the interval [1, +∞], which is the sine of the arctangent of the square root of x. By analyzing the image sets of square root, arctangent, and sine functions successively, the instructor deduces the image set of their composition. The lecture demonstrates the continuity and monotonicity of each function, leading to the conclusion that the image set of the composite function is the interval [√2/2, 1) closed on the left and open on the right.