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Dado o gráfico (completo) da função \(g\), responda às perguntas a seguir.
![[Picture]](/rendered/understandingFunctions/exercises/piecewise1-9746bc0ce13f0620fc0456c735a06d37.svg)
Determine o domínio de \(g\). \[ \left [\class{xm-answer}{\cssId{xm-answer-1}{\phantom{\rule{7em}{1.5em}}}},\class{xm-answer}{\cssId{xm-answer-2}{\phantom{\rule{7em}{1.5em}}}}\right ] \]
Determine a imagem de \(g\). \[ \left (\class{xm-answer}{\cssId{xm-answer-3}{\phantom{\rule{7em}{1.5em}}}},\class{xm-answer}{\cssId{xm-answer-4}{\phantom{\rule{7em}{1.5em}}}}\right ] \cup \left [\class{xm-answer}{\cssId{xm-answer-5}{\phantom{\rule{7em}{1.5em}}}},\class{xm-answer}{\cssId{xm-answer-6}{\phantom{\rule{7em}{1.5em}}}}\right ) \]
Encontre uma fórmula para \(g\).
xmhintNo intervalo \((-2,0]\), o gráfico da função \(g\) é uma reta. Portanto, nesse intervalo, \(g(x)=ax+b\),
para certas constantes \(a\) e \(b\). O mesmo vale para \(g(x)\) no intervalo \((0,2]\).
\[ g(x) = \begin{cases} \class{xm-answer}{\cssId{xm-answer-7}{\phantom{\rule{7em}{1.5em}}}} &\text {se $x=-2$}\\ \class{xm-answer}{\cssId{xm-answer-8}{\phantom{\rule{7em}{1.5em}}}} &\text {se $-2<x\le 0$}\\ \class{xm-answer}{\cssId{xm-answer-9}{\phantom{\rule{7em}{1.5em}}}}&\text {se $0<x\le 2$.} \end{cases} \]
Sejam \(f(x) = |x|\) e
\(h(x) = \frac {4x}{x+20}\).
Lembre-se: \[ f(x) = |x| = \begin{cases} -x & \text {se $x<0$,}\\ x & \text {se $0\le x$.} \end{cases} \]
Determine o domínio de \(g\circ f\). \[ \left [\class{xm-answer}{\cssId{xm-answer-10}{\phantom{\rule{7em}{1.5em}}}},\class{xm-answer}{\cssId{xm-answer-11}{\phantom{\rule{7em}{1.5em}}}}\right ] \]
Encontre uma fórmula para \(g\circ f\).
xmhintObservação: se
\(-2\le x\le 2\), então \(0\le f(x)\le 2\)
\[ g\circ f = \begin{cases} \class{xm-answer}{\cssId{xm-answer-12}{\phantom{\rule{7em}{1.5em}}}} & \text {se $x=0$,}\\ \class{xm-answer}{\cssId{xm-answer-13}{\phantom{\rule{7em}{1.5em}}}} &\text {se $-2<x<0$,}\\ \class{xm-answer}{\cssId{xm-answer-14}{\phantom{\rule{7em}{1.5em}}}}&\text {se $0<x\le 2$.} \end{cases} \]
Calcule (escreva um número ou “DNE”): \[ f(g(h(10))) = \class{xm-answer}{\cssId{xm-answer-15}{\phantom{\rule{7em}{1.5em}}}}. \]
Calcule (escreva
um número ou “DNE”): \[ f(g(h(-10))) = \class{xm-answer}{\cssId{xm-answer-16}{\phantom{\rule{7em}{1.5em}}}}. \]
Calcule \(h(f(-20))\).
\[ h(f(-20)) =h\left (\class{xm-answer}{\cssId{xm-answer-17}{\phantom{\rule{7em}{1.5em}}}} \right )=\frac {4\cdot \class{xm-answer}{\cssId{xm-answer-18}{\phantom{\rule{7em}{1.5em}}}}}{40}=\class{xm-answer}{\cssId{xm-answer-19}{\phantom{\rule{7em}{1.5em}}}} \]
Encontre uma expressão para \(h(f(x))\). \[ h(f(x)) = \begin{cases} h\left ( \class{xm-answer}{\cssId{xm-answer-20}{\phantom{\rule{7em}{1.5em}}}}\right ) & \text {se $x<0$,}\\ h\left ( \class{xm-answer}{\cssId{xm-answer-21}{\phantom{\rule{7em}{1.5em}}}}\right ) & \text {se $0\le x$.} \end{cases} \]
\[ h(f(x)) = \begin{cases} \class{xm-answer}{\cssId{xm-answer-22}{\phantom{\rule{7em}{1.5em}}}} & \text {if $x<0$,}\\ \\
\class{xm-answer}{\cssId{xm-answer-23}{\phantom{\rule{7em}{1.5em}}}} & \text {if $0\le x$.} \end{cases} \]