Let f:[- 2 , 2 ] , ,[-1,1]

MATHEMATICS · Subjective · AP · AP Precalculus · Trigonometric and Polar Functions · Trigonometric and Polar Functions

Problem

Let f:[−π2,π2] ⟶ [−1,1],f(x)=sin⁡x,f:[-\tfrac{\pi}{2},\tfrac{\pi}{2}]\,\longrightarrow\,[-1,1],\qquad f(x)=\sin x, g:[−1,1] ⟶ [−π2,π2],g(x)=arcsin⁡x.g:[-1,1]\,\longrightarrow\,[-\tfrac{\pi}{2},\tfrac{\pi}{2}],\qquad g(x)=\arcsin x. (a) Prove that f∘gf\circ g is the identity function on [−1,1][-1,1] and g∘fg\circ f is the identity function on [−π2,π2]\bigl[-\tfrac{\pi}{2},\tfrac{\pi}{2}\bigr]. (b) Explain why restricting the domains and codomains as shown is essential for ff and gg to be inverses.

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