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Trazado el gráfico de la función/ (4^cos(2*x)+4^(cos(x)^2)-3)/sqrt(5*x-2-2*x^2)

Gráfico de la función y = (4^cos(2*x)+4^(cos(x)^2)-3)/sqrt(5*x-2-2*x^2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                       2       
        cos(2*x)    cos (x)    
       4         + 4        - 3
f(x) = ------------------------
            ________________   
           /              2    
         \/  5*x - 2 - 2*x
f(x)=(4cos⁡2(x)+4cos⁡(2x))−3−2x2+(5x−2)f{\left(x \right)} = \frac{\left(4^{\cos^{2}{\left(x \right)}} + 4^{\cos{\left(2 x \right)}}\right) - 3}{\sqrt{- 2 x^{2} + \left(5 x - 2\right)}} 
f = (4^(cos(x)^2) + 4^cos(2*x) - 3)/sqrt(-2*x^2 + 5*x - 2)
Gráfico de la función
02468-8-6-4-2-1010-1010
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0.5x_{1} = 0.5
x2=2x_{2} = 2
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
(4cos⁡2(x)+4cos⁡(2x))−3−2x2+(5x−2)=0\frac{\left(4^{\cos^{2}{\left(x \right)}} + 4^{\cos{\left(2 x \right)}}\right) - 3}{\sqrt{- 2 x^{2} + \left(5 x - 2\right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=18.0641577581413x_{1} = 18.0641577581413
x2=−46.3384916404494x_{2} = -46.3384916404494
x3=52.621676947629x_{3} = 52.621676947629
x4=49.4800842940392x_{4} = 49.4800842940392
x5=93.4623814442964x_{5} = 93.4623814442964
x6=33.7721210260903x_{6} = 33.7721210260903
x7=38.484510006475x_{7} = 38.484510006475
x8=71.4712328691678x_{8} = 71.4712328691678
x9=−2.35619449019234x_{9} = -2.35619449019234
x10=−173.572994110836x_{10} = -173.572994110836
x11=−69.9004365423729x_{11} = -69.9004365423729
x12=32.2013246992954x_{12} = 32.2013246992954
x13=46.3384916404494x_{13} = 46.3384916404494
x14=60.4756585816035x_{14} = 60.4756585816035
x15=−98.174770424681x_{15} = -98.174770424681
x16=−82.4668071567321x_{16} = -82.4668071567321
x17=30.6305283725005x_{17} = 30.6305283725005
x18=−91.8915851175014x_{18} = -91.8915851175014
x19=47.9092879672443x_{19} = 47.9092879672443
x20=54.1924732744239x_{20} = 54.1924732744239
x21=19.6349540849362x_{21} = 19.6349540849362
x22=11.7809724509617x_{22} = 11.7809724509617
x23=69.9004365423729x_{23} = 69.9004365423729
x24=−102.887159405066x_{24} = -102.887159405066
x25=−19.6349540849362x_{25} = -19.6349540849362
x26=68.329640215578x_{26} = 68.329640215578
x27=−21.2057504117311x_{27} = -21.2057504117311
x28=377.776516594173x_{28} = 377.776516594173
x29=−57.3340659280137x_{29} = -57.3340659280137
x30=−24.3473430653209x_{30} = -24.3473430653209
x31=−85.6083998103219x_{31} = -85.6083998103219
x32=−47.9092879672443x_{32} = -47.9092879672443
x33=16.4933614313464x_{33} = 16.4933614313464
x34=−32.2013246992954x_{34} = -32.2013246992954
x35=25.9181393921158x_{35} = 25.9181393921158
x36=40.0553063332699x_{36} = 40.0553063332699
x37=−33.7721210260903x_{37} = -33.7721210260903
x38=44.7676953136546x_{38} = 44.7676953136546
x39=8.63937979737193x_{39} = 8.63937979737193
x40=27.4889357189107x_{40} = 27.4889357189107
x41=−79.3252145031423x_{41} = -79.3252145031423
x42=24.3473430653209x_{42} = 24.3473430653209
x43=−25.9181393921158x_{43} = -25.9181393921158
x44=22.776546738526x_{44} = 22.776546738526
x45=−13.3517687777566x_{45} = -13.3517687777566
x46=−16.4933614313464x_{46} = -16.4933614313464
x47=−62.0464549083984x_{47} = -62.0464549083984
x48=−49.4800842940392x_{48} = -49.4800842940392
x49=−5.49778714378214x_{49} = -5.49778714378214
x50=10.2101761241668x_{50} = 10.2101761241668
x51=76.1836218495525x_{51} = 76.1836218495525
x52=99.7455667514759x_{52} = 99.7455667514759
x53=90.3207887907066x_{53} = 90.3207887907066
x54=−74.6128255227576x_{54} = -74.6128255227576
x55=−35.3429173528852x_{55} = -35.3429173528852
x56=−3.92699081698724x_{56} = -3.92699081698724
x57=−55.7632696012188x_{57} = -55.7632696012188
x58=−93.4623814442964x_{58} = -93.4623814442964
x59=−40.0553063332699x_{59} = -40.0553063332699
x60=−0.785398163397448x_{60} = -0.785398163397448
x61=−68.329640215578x_{61} = -68.329640215578
x62=77.7544181763474x_{62} = 77.7544181763474
x63=91.8915851175014x_{63} = 91.8915851175014
x64=−99.7455667514759x_{64} = -99.7455667514759
x65=66.7588438887831x_{65} = 66.7588438887831
x66=−38.484510006475x_{66} = -38.484510006475
x67=55.7632696012188x_{67} = 55.7632696012188
x68=−54.1924732744239x_{68} = -54.1924732744239
x69=−41.6261026600648x_{69} = -41.6261026600648
x70=−178.285383091221x_{70} = -178.285383091221
x71=−90.3207887907066x_{71} = -90.3207887907066
x72=−76.1836218495525x_{72} = -76.1836218495525
x73=−77.7544181763474x_{73} = -77.7544181763474
x74=3.92699081698724x_{74} = 3.92699081698724
x75=5.49778714378214x_{75} = 5.49778714378214
x76=−18.0641577581413x_{76} = -18.0641577581413
x77=−825.453469730718x_{77} = -825.453469730718
x78=−84.037603483527x_{78} = -84.037603483527
x79=−11.7809724509617x_{79} = -11.7809724509617
x80=88.7499924639117x_{80} = 88.7499924639117
x81=74.6128255227576x_{81} = 74.6128255227576
x82=−43.1968989868597x_{82} = -43.1968989868597
x83=−10.2101761241668x_{83} = -10.2101761241668
x84=−71.4712328691678x_{84} = -71.4712328691678
x85=−27.4889357189107x_{85} = -27.4889357189107
x86=82.4668071567321x_{86} = 82.4668071567321
x87=63.6172512351933x_{87} = 63.6172512351933
x88=−121.736715326604x_{88} = -121.736715326604
x89=62.0464549083984x_{89} = 62.0464549083984
x90=85.6083998103219x_{90} = 85.6083998103219
x91=41.6261026600648x_{91} = 41.6261026600648
x92=98.174770424681x_{92} = 98.174770424681
x93=96.6039740978861x_{93} = 96.6039740978861
x94=−63.6172512351933x_{94} = -63.6172512351933
x95=84.037603483527x_{95} = 84.037603483527
x96=−60.4756585816035x_{96} = -60.4756585816035
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en (4^cos(2*x) + 4^(cos(x)^2) - 3)/sqrt(5*x - 2 - 2*x^2).
−3+(4cos⁡(0⋅2)+4cos⁡2(0))(−2+0⋅5)−2⋅02\frac{-3 + \left(4^{\cos{\left(0 \cdot 2 \right)}} + 4^{\cos^{2}{\left(0 \right)}}\right)}{\sqrt{\left(-2 + 0 \cdot 5\right) - 2 \cdot 0^{2}}}
Resultado:
f(0)=−52i2f{\left(0 \right)} = - \frac{5 \sqrt{2} i}{2}
Punto:
(0, -5*i*sqrt(2)/2)
Asíntotas verticales
Hay:
x1=0.5x_{1} = 0.5
x2=2x_{2} = 2
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((4cos⁡2(x)+4cos⁡(2x))−3−2x2+(5x−2))=0\lim_{x \to -\infty}\left(\frac{\left(4^{\cos^{2}{\left(x \right)}} + 4^{\cos{\left(2 x \right)}}\right) - 3}{\sqrt{- 2 x^{2} + \left(5 x - 2\right)}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞((4cos⁡2(x)+4cos⁡(2x))−3−2x2+(5x−2))=0\lim_{x \to \infty}\left(\frac{\left(4^{\cos^{2}{\left(x \right)}} + 4^{\cos{\left(2 x \right)}}\right) - 3}{\sqrt{- 2 x^{2} + \left(5 x - 2\right)}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (4^cos(2*x) + 4^(cos(x)^2) - 3)/sqrt(5*x - 2 - 2*x^2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((4cos⁡2(x)+4cos⁡(2x))−3x−2x2+(5x−2))=0\lim_{x \to -\infty}\left(\frac{\left(4^{\cos^{2}{\left(x \right)}} + 4^{\cos{\left(2 x \right)}}\right) - 3}{x \sqrt{- 2 x^{2} + \left(5 x - 2\right)}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((4cos⁡2(x)+4cos⁡(2x))−3x−2x2+(5x−2))=0\lim_{x \to \infty}\left(\frac{\left(4^{\cos^{2}{\left(x \right)}} + 4^{\cos{\left(2 x \right)}}\right) - 3}{x \sqrt{- 2 x^{2} + \left(5 x - 2\right)}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
(4cos⁡2(x)+4cos⁡(2x))−3−2x2+(5x−2)=(4cos⁡2(x)+4cos⁡(2x))−3−2x2−5x−2\frac{\left(4^{\cos^{2}{\left(x \right)}} + 4^{\cos{\left(2 x \right)}}\right) - 3}{\sqrt{- 2 x^{2} + \left(5 x - 2\right)}} = \frac{\left(4^{\cos^{2}{\left(x \right)}} + 4^{\cos{\left(2 x \right)}}\right) - 3}{\sqrt{- 2 x^{2} - 5 x - 2}}
- No
(4cos⁡2(x)+4cos⁡(2x))−3−2x2+(5x−2)=−(4cos⁡2(x)+4cos⁡(2x))−3−2x2−5x−2\frac{\left(4^{\cos^{2}{\left(x \right)}} + 4^{\cos{\left(2 x \right)}}\right) - 3}{\sqrt{- 2 x^{2} + \left(5 x - 2\right)}} = - \frac{\left(4^{\cos^{2}{\left(x \right)}} + 4^{\cos{\left(2 x \right)}}\right) - 3}{\sqrt{- 2 x^{2} - 5 x - 2}}
- No
es decir, función
no es
par ni impar
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