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Thread: Range of f(x)?

  1. #1
    Super Member fardeen_gen's Avatar
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    Range of f(x)?

    Given the functions
    $\displaystyle f(x) = \exp(\arccos \left(\sin \left(x + \frac{\pi}{3}\right)\right))$,
    $\displaystyle g(x) = \mbox{arccsc} \left(\frac{4 - 2\cos x}{3}\right)$
    and the function $\displaystyle h(x) = f(x)$ defined only for the common domain of $\displaystyle f(x)\ \mbox{\&}\ g(x)$.

    Calculate range of $\displaystyle h(x)$.
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  2. #2
    Senior Member
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    Since sines and cosines take on the same values,

    $\displaystyle f(x)=\exp\left(\arccos\left(\sin\left(x+\frac{\pi} {3}\right)\right)\right)$

    is defined everywhere, i.e., its domain is $\displaystyle \mathbb{R}$. The domain of $\displaystyle h$ is therefore just the domain of $\displaystyle g$, consisting of precisely the values of $\displaystyle x$ for which

    $\displaystyle \frac{4-2\cos x}{3}$

    belongs to $\displaystyle (-\infty,-1]\cup[1,\infty)$. When we find these values, we can use them to find the range of $\displaystyle h$ on that domain.
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  3. #3
    Super Member fardeen_gen's Avatar
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    How do we find these values?
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