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6.3: Ejercicios

Última actualización

30 oct 2022
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- 6.2: Operaciones sobre funciones dadas por tablas
- 7: La inversa de una función

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

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Ejercicio $\PageIndex{1}$

Encuentre $f+g$ , $f-g$ , $f\cdot g$ para las funciones a continuación. Declarar su dominio.

$f(x)=x^2+6x$ , $\quad \quad$ y $g(x)=3x-5$
$f(x)=x^3+5$ , $\quad \quad$ y $g(x)=5x^2+7$
$f(x)=3x+7\sqrt{x}$ , $\quad$ y $g(x)=2x^2+5\sqrt{x}$
$f(x)=\dfrac{1}{x+2}$ , $\quad$ y $g(x)=\dfrac{5x}{x+2}$
$f(x)=\sqrt{x-3}$ , $\quad \quad$ y $g(x)=2\sqrt{x-3}$
$f(x)=x^2+2x+5$ , $\quad \quad$ y $g(x)=3x-6$
$f(x)=x^2+3x$ , $\quad \quad$ y $g(x)=2x^2+3x+4$

Contestar

$(f+g)(x)=x^{2}+9 x-5$ con dominio $D_{f+g}=\mathbb{R},(f-g)(x)= x^2 + 3x + 5$ con dominio $D_{f-g}=\mathbb{R},(f \cdot g)(x)=3 x^{3}+13 x^{2}-30 x$ con dominio $D_{f \cdot g}=\mathbb{R}$
$(f+g)(x)=x^{3}+5 x^{2}+12, D_{f+g}=\mathbb{R},(f-g)(x)=x^{3}-5 x^{2}-2, D_{f-g}=\mathbb{R},(f \cdot g)(x)=5 x^{5}+7 x^{3}+25 x^{2}+35, D_{f \cdot g}=\mathbb{R}$
$(f+g)(x)=2 x^{2}+3 x+12 \sqrt{x}, D_{f+g}=[0, \infty),(f-g)(x)=-2 x^{2}+3 x+2 \sqrt{x}, D_{f-g}=[0, \infty),(f \cdot g)(x)=6 x^{3}+14 x^{2} \sqrt{x}+15 x \sqrt{x}+35 x, D_{f \cdot g}=[0, \infty)$
$(f+g)(x)=\dfrac{5 x+1}{x+2}, D_{f+g}=\mathbb{R}-\{-2\},(f-g)(x)=\dfrac{1-5 x}{x+2}, D_{f-g}=\mathbb{R}-\{-2\},(f \cdot g)(x)=\dfrac{5 x}{(x+2)^{2}}, D_{f \cdot g}=\mathbb{R}-\{-2\}$
$(f+g)(x)=3 \sqrt{x-3}, D_{f+g}=[3, \infty),(f-g)(x)=-\sqrt{x-3}, D_{f-g}=[3, \infty),(f \cdot g)(x)=2 \cdot(\sqrt{x-3})^{2}=2 \cdot(x-3), D_{f \cdot g}=[3, \infty)$
$(f+g)(x)=x^{2}+5 x-1, D_{f+g}=\mathbb{R},(f-g)(x)=x^{2}-x+11, D_{f-g}=\mathbb{R},(f \cdot g)(x)=3 x^{3}+3 x-30, D_{f \cdot g}=\mathbb{R}$
$(f+g)(x)=3 x^{2}+6 x+4, D_{f+g}=\mathbb{R},(f-g)(x)=-x^{2}-4, D_{f-g}=\mathbb{R},(f \cdot g)(x)=2 x^{4}+9 x^{3}+13 x^{2}+12 x, D_{f \cdot g}=\mathbb{R}$

Ejercicio $\PageIndex{2}$

Encuentre $\dfrac f g$ , y $\dfrac g f$ para las funciones a continuación. Declarar su dominio.

$f(x)=3x+6$ , $\quad \quad$ y $g(x)=2x-8$
$f(x)=x+2$ , $\quad \quad$ y $g(x)=x^2-5x+4$
$f(x)=\dfrac{1}{x-5}$ , $\quad \quad$ y $g(x)=\dfrac{x-2}{x+3}$
$f(x)=\sqrt{x+6}$ , $\quad \quad$ y $g(x)=2x+5$
$f(x)=x^2+8x-33$ , $\quad \quad$ y $g(x)=\sqrt{x}$

Contestar

$\left(\dfrac{f}{g}\right)(x)=\dfrac{3 x+6}{2 x-8}$ con dominio $D_{\frac{f}{g}}=\mathbb{R}-\{4\}$ , $\left(\dfrac{g}{f}\right)(x)=\dfrac{2 x-8}{3 x+6}$ con dominio $D_{\frac{g}{f}}=\mathbb{R}-\{-2\}$
$\left(\dfrac{f}{g}\right)(x)=\dfrac{x+2}{x^{2}-5 x+4}=\dfrac{x+2}{(x-4)(x-1)}$ , $D_{\frac{f}{g}}=\mathbb{R}-\{1,4\}, \quad\left(\dfrac{g}{f}\right)(x)=\dfrac{x^{2}-5 x+4}{x+2}, D_{\frac{g}{f}}=\mathbb{R}-\{-2\}$
$\left(\dfrac{f}{g}\right)(x)=\dfrac{x+3}{(x-5)(x-2)}, D_{\frac{f}{g}}=\mathbb{R}-\{-3,2,5\},\left(\dfrac{g}{f}\right)(x)=\dfrac{(x-5)(x-2)}{x+3}, D_{\frac{g}{f}}=\mathbb{R}-\{-3,5\}$
$\left(\dfrac{f}{g}\right)(x)=\dfrac{\sqrt{x+6}}{2 x+5}, D_{\frac{f}{g}}=\left[-6,-\dfrac{5}{2}\right) \cup\left(-\dfrac{5}{2}, \infty\right), \left(\dfrac{g}{f}\right)(x)=\dfrac{2 x+5}{\sqrt{x+6}}, D_{\frac{g}{f}}=(-6, \infty)$
$\left(\dfrac{f}{g}\right)(x)=\dfrac{x^{2}+8 x-33}{\sqrt{x}}, D_{\frac{f}{g}}=(0, \infty), \left(\dfrac{g}{f}\right)(x)=\dfrac{\sqrt{x}}{x^{2}+8 x-33}, D_{\frac{g}{f}}=[0,3) \cup(3, \infty)$

Ejercicio $\PageIndex{3}$

Dejar $f(x)=2x-3$ y $g(x)=3x^2+4x$ . Encuentra las siguientes composiciones

$f(g(2))$
$g(f(2))$
$f(f(5))$
$f(5 g(-3))$
$g(f(2)-2)$
$f(f(3)+g(3))$
$g(f(2+x))$
$f(f(-x))$
$f( f(-3)-3 g(2))$
$f(f(f(2)))$
$f(x+h)$
$g(x+h)$

Contestar

$37$
$7$
$11$
$147$
$-1$
$81$
$12 x^{2}+20 x+7$
$-4 x-9$
$-141$
$-5$
$2 x+2 h-3$
$3 x^{2}+6 x h+3 h^{2}+4 x+4 h$

Ejercicio $\PageIndex{4}$

Encuentra la composición $(f\circ g)(x)$ para las funciones:

$f(x)=3x-5$ , $\quad \quad$ y $g(x)=2x+3$
$f(x)=x^2+2$ , $\quad \quad$ y $g(x)=x+3$
$f(x)=x^2-3x+2$ , $\quad \quad$ y $g(x)=2x+1$
$f(x)=x^2+\sqrt{x+3}$ , $\quad \quad$ y $g(x)=x^2+2x$
$f(x)=\dfrac{2}{x+4}$ , $\quad \quad$ y $g(x)=x+h$
$f(x)=x^2+4x+3$ , $\quad \quad$ y $g(x)=x+h$

Contestar

$(f \circ g)(x)=6 x+4$
$(f \circ g)(x)=x^{2}+6 x+11$
$(f \circ g)(x)=4 x^{2}-2x$
$(f \circ g)(x)=x^{4}+4 x^{3}+4 x^{2}+\sqrt{x^{2}+2 x+3}$
$(f \circ g)(x)=\dfrac{2}{x+h+4}$
$(f \circ g)(x)=x^{2}+2 x h+h^{2}+4 x+4 h+3$

Ejercicio $\PageIndex{5}$

Encuentra las composiciones

$(f\circ g)(x),\quad (g\circ f)(x),\quad(f\circ f)(x),\quad(g\circ g)(x) \nonumber$

para las siguientes funciones:

$f(x)=2x+4$ , $\quad \quad$ y $g(x)=x-5$
$f(x)=x+3$ , $\quad \quad$ y $g(x)=x^2-2x$
$f(x)=2x^2-x-6$ , $\quad \quad$ y $g(x)=\sqrt{3x+2}$
$f(x)=\dfrac{1}{x+3}$ , $\quad \quad$ y $g(x)=\dfrac{1}{x}-3$
$f(x)=(2x-7)^2$ , $\quad \quad$ y $g(x)=\dfrac{\sqrt{x}+7}{2}$

Contestar

$(f \circ g)(x)=2 x-6, (g \circ f)(x)=2 x-1, (f \circ f)(x)=4 x+12, (g \circ g)(x)=x-10$
$(f \circ g)(x)=x^{2}-2 x+3, (g \circ f)(x)=x^{2}+4 x+3, (f \circ f)(x)=x+6, (g \circ g)(x)=x^{4}-4 x^{3}+2 x^{2}+4 x$
$(f \circ g)(x)=6 x-2-\sqrt{3 x+2}, (g \circ f)(x)=\sqrt{6 x^{2}-3 x-16}, (f \circ f)(x)=8 x^{4}-8 x^{3}-48 x^{2}+25 x+72, (g \circ g)(x)=\sqrt{3 \sqrt{3 x+2}+2}$
$(f \circ g)(x)=x, (g \circ f)(x)=x, (f \circ f)(x)=\dfrac{x+3}{3 x+10}, (g \circ g)(x)=\dfrac{10 x-3}{1-3 x}$
$(f \circ g)(x)=x,(g \circ f)(x)=x,(f \circ f)(x)=\left(2(2 x-7)^{2}-7\right)^{2}$ o expandido en grados decrecientes: $(f \circ f)(x)=64 x^{4}-896 x^{3}+4592 x^{2}-10192 x+8281,(g \circ g)(x)=\dfrac{\sqrt{\frac{\sqrt{x}+7}{2}}+7}{2}=\dfrac{14+\sqrt{14+2 \sqrt{x}}}{4}$

Ejercicio $\PageIndex{6}$

Dejar $f$ y $g$ ser las funciones definidas por la siguiente tabla. Completa la tabla que figura a continuación.

\ [\ begin {array} {|c||c|c|c|c|c|c|c|c|}
\ hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7\
\ hline\ hline\ hline f (x) & 4 & 5 & 7 & 0 & -2 & 6 & 4
\\ hline g (x) & 6 & -8 & 5 & 2 & 9 & 11 & 2\\
\ hline f (x) +3 & & & & & & & &\
\ hline 4 g (x) +5 & & & & & & & & & &
\ hline g (x) -2 f (x) & & & & & & & & & & & & & &
\ hline f (x+3) & & & & & & &\
\ hline
\ end {array}\ nonumber\]

Contestar

\ (\ begin {array} {|c||c|c|c|c|c|c|c|c|}
\ hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7\
\ hline\ hline\ hline f (x) & 4 & 5 & 7 & 0 & -2 & 6 & 4
\\ hline g (x) & 6 & -8 & 5 & 2 & 9 & 11 & 2\\
\ hline f (x) +3 & 7 & 8 & 10 & 3 & 1 & 9 & 7\
\ hline 4 g (x) +5 & 29 & -28 & 25 & 13 & 41 & 49 & 13\
\ hline g (x) -2 f (x) & -2 & -18 & -9 & 2 & 13 & -1 y -6\\
\ hline f (x+3) & 0 & -2 & 6 & 4 &\ text {undef.} &\ text {undef.} &\ text {undef.}\\
\ hline
\ end {array}\ nonumber\)

Tenga en cuenta, sin embargo, que la tabla completa para $y = f(x + 3)$ viene dada por:

\ (\ begin {array} {|c||c|c|c|c|c|c|c|c|}
\ hline x & -2 & -1 & 0 & 1 & 2 & 3 & 4\
\ hline\ hline\ hline f (x+3) & 4 & 5 & 7 & 0 & -2 & 6 & 4\
\ hline
\ end {array}\ nonumber\)

Ejercicio $\PageIndex{7}$

Dejar $f$ y $g$ ser las funciones definidas por la siguiente tabla. Completa la tabla componiendo las funciones dadas.

\ [\ begin {array} {|c||c|c|c|c|c|c|}
\ hline x & 1 & 2 & 3 & 4 & 5 & 6\
\ hline\ hline\ hline f (x) & 3 & 1 & 2 & 5 & 6 & 3
\\ hline g (x) & 5 & 2 & 6 & 1 & 2 & 4\
\ hline (g\ circ f) (x) y & & & &\
\ hline (f\ circ g) (x) & & & & & & &\
\ hline (f\ circ f) (x) & & & & & & &\
\ hline (g\ circ g) (x) & & & & &\
\ hline
\\ final {matriz}\ nonumber\]

Contestar: \ (\ begin {array} {|c||c|c|c|c|c|c|}
\ hline x & 1 & 2 & 3 & 4 & 5 & 6\
\ hline\ hline\ hline f (x) & 3 & 1 & 2 & 5 & 6 & 3
\\ hline g (x) & 5 & 2 & 6 & 1 & 2 & 4\
\ hline (g\ circ f) (x) y 6 y 5 y 2 y 2 y 4 y 6\
\ hline (f\ circ g) (x) y 6 y 1 y 3 y 3 y 1 y 5\
\ hline (f\ circ f) (x) y 2 y 3 y 1 y 6 y 3 y 2\
\ hline (g\ circ g) (x) y 2 y 2 y 4 & 5 y 2 y 1\\
\ hline
\ end {array}\ nonumber\)

Ejercicio $\PageIndex{8}$

Dejar $f$ y $g$ ser las funciones definidas por la siguiente tabla. Completa la tabla componiendo las funciones dadas.

\ [\ begin {array} {|c||c|c|c|c|c|c|c|c|}
\ hline x & 0 & 2 & 4 & 6 & 8 & 10 & 12\
\ hline\ hline\ hline f (x) & 4 & 8 & 5 & 6 & 12 & -1 & 10\
\ hline g (x) & 10 & 2 & 0 & 0 & 6 & 7 & 2 & 8\\
\ hline (g\ circ f) (x) & & & & & & &\
\ hline (f\ circ g) (x) & & & & & & & &
\\ hline (f\ circ f) (x) & & & & & &\
\ hline (g\ circ g) (x) & & & & & ; &\\
\ hline
\ end {array}\ nonumber\]

Contestar: \ (\ begin {array} {|c||c|c|c|c|c|c|c|c|}
\ hline x & 0 & 2 & 4 & 6 & 8 & 10 & 10 & 12\
\ hline\ hline\ hline f (x) & 4 & 8 & 5 & 6 & 12 & -1 & 10
\\ hline g (x) & 10 & 2 & 0 & 6 & 7 & 2 & 8 \
\ hline (g\ circ f) (x) & 0 & 7 &\ text {undef.} & -6 & 8 &\ text {undef.} & 2\\
\ hline (f\ circ g) (x) & -1 & 8 & 4 &\ text {undef.} &\ text {undef.} & 8 & 12
\\ hline (f\ circ f) (x) y 5 amp; 12 &\ text {undef.} & 6 & 10 &\ text {undef.} & -1\\
\ hline (g\ circ g) (x) & 2 & 2 & 10 &\ text {undef.} &\ text {undef.} & 2 & 7\
\ hline
\ end {array}\ nonumber\)

Ejercicio6.3.1\PageIndex{1}

Ejercicio6.3.2\PageIndex{2}

Ejercicio6.3.3\PageIndex{3}

Ejercicio6.3.4\PageIndex{4}

Ejercicio6.3.5\PageIndex{5}

Ejercicio6.3.6\PageIndex{6}

Ejercicio6.3.7\PageIndex{7}

Ejercicio6.3.8\PageIndex{8}

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Ejercicio $\PageIndex{1}$

Ejercicio $\PageIndex{2}$

Ejercicio $\PageIndex{3}$

Ejercicio $\PageIndex{4}$

Ejercicio $\PageIndex{5}$

Ejercicio $\PageIndex{6}$

Ejercicio $\PageIndex{7}$

Ejercicio $\PageIndex{8}$