Categoria: Função inversa

0

A inversa de uma função

Função inversa: Aleph 11 - Volume 2 Pág. 174 Ex. 15

Enunciado

A função $f$ tem domínio $\left[ {0, + \infty } \right[$ e é definida por $f\left( x \right) = 4{x^2} + 1$.

  1. Esboce o gráfico de $f$ e indique o contradomínio da função.
     
  2. Explique porque existe inversa de $f$ e determine uma expressão para ${f^{ – 1}}\left( x \right)$.
Caracterize a função inversa 0

Caracterize a função inversa

Função inversa: Aleph 11 - Volume 2 Pág. 173 Ex. 8

Enunciado

Caracterize a função inversa de cada uma das seguintes funções:

\[\begin{array}{*{20}{c}}
{f\left( x \right) = 6x + 5}&{}&{}&{g\left( x \right) =  – \frac{{12}}{{x + 3}}}
\end{array}\]

Resolução >> Resolução

\[\begin{array}{*{20}{c}}
{f\left( x \right) = 6x + 5}&{}&{}&{g\left( x \right) =  – \frac{{12}}{{x + 3}}}
\end{array}\]

Como ${D_f} = D{‘_f} = \mathbb{R}$ e $y = 6x + 5 \Leftrightarrow x = \frac{{y – 5}}{6}$, então: \[\begin{array}{*{20}{l}}
{{f^{ – 1}}:}&{\mathbb{R} \to \mathbb{R}} \\
{}&{x \to \frac{x}{6} – \frac{5}{6}}
\end{array}\]

Como ${D_g} = \mathbb{R}\backslash \left\{ { – 3} \right\}$, $D{‘_g} = \mathbb{R}\backslash \left\{ 0 \right\}$ e $y =  – \frac{{12}}{{x + 3}} \Leftrightarrow x =  – 3 – \frac{{12}}{y}$, então: \[\begin{array}{*{20}{l}}
{{g^{ – 1}}:}&{\mathbb{R}\backslash \left\{ 0 \right\} \to \mathbb{R}} \\
{}&{x \to  – 3 – \frac{{12}}{x}}
\end{array}\]

 

var parameters = { "id": "ggbApplet", "width":760, "height":565, "showMenuBar":false, "showAlgebraInput":false, "showToolBar":false, "customToolBar":"0 39 59 || 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 || 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 || 36 46 , 38 49 50 , 71 | 30 29 54 32 31 33 | 17 26 62 , 14 66 68 | 25 52 60 61 || 40 41 42 , 27 28 35 , 6", "showToolBarHelp":false, "showResetIcon":true, "enableLabelDrags":false, "enableShiftDragZoom":false, "enableRightClick":false, "errorDialogsActive":false, "useBrowserForJS":false, "preventFocus":false, "language":"pt", // use this instead of ggbBase64 to load a material from GeoGebraTube // "material_id":12345, "ggbBase64":"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"}; // is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View var views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 0,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0}; var applet = new GGBApplet(parameters, '5.0', views); window.onload = function() {applet.inject('ggbApplet')};

<< Enunciado
0

Use a calculadora gráfica

Função inversa: Infinito 11 A - Parte 2 Pág. 204 Ex. 75

Enunciado

  1. Use a calculadora gráfica e conjecture quais das seguintes funções polinomiais têm função inversa:
     
    ${{f}_{1}}(x)={{x}^{3}}-9{{x}^{2}}+5x-5$ ${{f}_{2}}(x)=2x+{{x}^{2}}$ ${{f}_{3}}(x)={{x}^{3}}-2{{x}^{2}}+5x-5$
    ${{f}_{4}}(x)=2x-{{x}^{2}}$ ${{f}_{5}}(x)=2x+{{x}^{3}}$ ${{f}_{6}}(x)=2x-{{x}^{3}}$

     

  2. Para as que admitiu serem funções injectivas, calcule a imagem, pela inversa, de 10, com aproximação às centésimas.

Resolução >> Resolução

  1.  
    ${{f}_{1}}(x)={{x}^{3}}-9{{x}^{2}}+5x-5$
    A função não admite inversa, pois é uma função não injectiva.
0

Qual o valor lógico das proposições?

Função inversa: Infinito 11 A - Parte 2 Pág. 204 Ex. 74

Enunciado

Qual o valor lógico das proposições?

  1. A função $f:x\to {{x}^{2}}-2$ admite função inversa.
     
  2. Nenhuma função par admite função inversa.
     
  3. Algumas funções ímpares admitem função inversa.

Resolução >> Resolução

  1. A afirmação é falsa.
     
    A função $f:x\to {{x}^{2}}-2$ não admite função inversa, pois não é uma função injectiva.
     
    Com efeito, é falsa a proposição ${{x}_{1}}\ne {{x}_{2}}\Rightarrow f({{x}_{1}})\ne f({{x}_{2}}),\forall {{x}_{1}},{{x}_{2}}\in {{D}_{f}}$, já que, por exemplo, $f(-1)=f(1)=-1$.
Caracterize a função inversa 1

Caracterize a função inversa

Função inversa: Infinito 11 A - Parte 2 Pág. 203 Ex. 72

Enunciado

Caracterize a função inversa das seguintes funções de variável real:

  1. $x\to f(x)=3x+2$
     
  2. $x\to g(x)=\frac{2-x}{x}$
     
  3. $x\to h(x)=\frac{x-5}{x+2}$
     
  4. $x\to i(x)={{x}^{3}}-3$

Resolução >> Resolução

  1. Ora, ${{D}_{f}}=\mathbb{R}$ e ${{D}_{f}}’=\mathbb{R}$.
    \[y=3x+2\Leftrightarrow 3x=y-2\Leftrightarrow x=\frac{1}{3}y-\frac{2}{3}\]
    Logo, \[\begin{array}{*{35}{l}}
       {{f}^{-1}}: & \mathbb{R}\to \mathbb{R}  \\
       {} & x\to \frac{1}{3}x-\frac{2}{3}  \\
    \end{array}\]
     
  2. Ora, ${{D}_{g}}=\mathbb{R}\backslash \left\{ 0 \right\}$ e ${{D}_{g}}’=\mathbb{R}\backslash \left\{ -1 \right\}$ (note que $g(x)=\frac{2-x}{x}=-1+\frac{2}{x}$).