For functions f and g,where f: [0, frac{pi}{2 | vedclass

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(A) Let $h(x) = (f+g)(x) = \sin x + \cos x = \sqrt{2} \sin(x + \frac{\pi}{4})$.For $x \in [0, \frac{\pi}{2}]$,$x + \frac{\pi}{4} \in [\frac{\pi}{4}, \

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$f+g$ is not one-one and $fg$ is not one-one

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(A) Let $h(x) = (f+g)(x) = \sin x + \cos x = \sqrt{2} \sin(x + \frac{\pi}{4})$.For $x \in [0, \frac{\pi}{2}]$,$x + \frac{\pi}{4} \in [\frac{\pi}{4}, \frac{3\pi}{4}]$.In this interval,the sine function is not monotonic (it increases then decreases),so $f+g$ is not one-one.Let $k(x) = (fg)(x) = \sin x \cos x = \frac{1}{2} \sin(2x)$.For $x \in [0, \frac{\pi}{2}]$,$2x \in [0, \pi]$.In this interval,the sine function is not monotonic (it increases then decreases),so $fg$ is not one-one.Thus,both $f+g$ and $fg$ are not one-one.

For functions $f$ and $g$,where $f: [0, \frac{\pi}{2}] \rightarrow R$ with $f(x) = \sin x$ and $g: [0, \frac{\pi}{2}] \rightarrow R$ with $g(x) = \cos x$,which of the following is true?

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