Mathematics

Mathematics

Problem

Identify the interval that can be taken as the domain of g(x)=sin⁡xg(x)=\sin x so that gg is invertible with codomain [−1,1][-1,1]. a) [0,π][0,\pi] b) [−π,π][-\pi,\pi] c) [−π2,π2]\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right] d) [−π4,3π4]\left[-\dfrac{\pi}{4},\dfrac{3\pi}{4}\right]
Intuitive explanation

For a function to have an inverse, it must be one-to-one (injective) on its domain. Since sin⁡x\sin x oscillates and repeats, you need to restrict it to an interval where it's strictly monotonic—meaning it never repeats a yy-value. The key insight is choosing an interval where sin⁡x\sin x covers the entire range [−1,1][-1,1] exactly once while increasing or decreasing throughout. This ensures a perfect one-to-one correspondence between inputs and outputs, making inversion possible.

Intuitive explanation

For a function to have an inverse, it must be one-to-one (injective) on its domain. Since sin⁡x\sin x oscillates and repeats, you need to restrict it to an interval where it's strictly monotonic—meaning it never repeats a yy-value. The key insight is choosing an interval where sin⁡x\sin x covers the entire range [−1,1][-1,1] exactly once while increasing or decreasing throughout. This ensures a perfect one-to-one correspondence between inputs and outputs, making inversion possible.

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