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Encontre uma expressão para \(f^{-1}(x)\), em que \(f^{-1}\) é a função inversa de \(f\).
xmhint\(y=1-\sin {\Bigl (\dfrac {\pi x}{2}\Bigr )}\)Troque \(x\) e \(y\) de lugar!
xmhint\[ \sin {\left (\dfrac {\pi y}{2}\right )}=1-x \]Isole \(y\)!
\[ f^{-1}(x)=\class{xm-answer}{\cssId{xm-answer-1}{\phantom{\rule{7em}{1.5em}}}} \]
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Qual é o domínio de \(f^{-1}\), a inversa de \(f\)?
xmhintLembre-se: domínio de \(f^{-1}\)=imagem de \(f\)!xmhintDetermine a imagem de \(f\). Lembre-se de que \[ -1\le -\sin {\Bigl (\dfrac {\pi x}{2}\Bigr )}\le 1 \] Somando \(1\), obtemos \[ 0\le \underbrace {1-\sin {\Bigl (\dfrac {\pi x}{2}\Bigr )}}_{\color {red}{f(x)}}\le 2 \]O domínio de \(f^{-1}=\left [\class{xm-answer}{\cssId{xm-answer-2}{\phantom{\rule{7em}{1.5em}}}},\class{xm-answer}{\cssId{xm-answer-3}{\phantom{\rule{7em}{1.5em}}}}\right ]\) -
Encontre uma expressão para \(\dfrac {d}{dx}f^{-1}(x)\), a derivada de \(f^{-1}\). \[ \ddx f^{-1}(x)=\class{xm-answer}{\cssId{xm-answer-4}{\phantom{\rule{7em}{1.5em}}}} \]
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Determine o valor de \(a\) para que a equação abaixo seja verdadeira. \[ \eval {\ddx f^{-1}(x)}_{x=1}=\frac {1}{f'(a)} \] \[ a=f^{-1}\left (\class{xm-answer}{\cssId{xm-answer-5}{\phantom{\rule{7em}{1.5em}}}}\right )=\frac {2}{\pi }\arcsin \left (\class{xm-answer}{\cssId{xm-answer-6}{\phantom{\rule{7em}{1.5em}}}}\right )=\class{xm-answer}{\cssId{xm-answer-7}{\phantom{\rule{7em}{1.5em}}}} \]
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Seja \(f\) a função dada por \[ f(x)=1-\sin {\Bigl (\dfrac {\pi x}{2}\Bigr )}, \quad \text {para }-1\le x \le 1. \]