operaciones con funciones (tarea prepanet mate 4)Descripción completa
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Gobiernos Radicales 1916-130Descripción completa
Descripción: EVALUACION HISTORIA NM2
Funciones racionale racionales s y con radicales
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Al igual que los polinomios se puede elaborar la grafica de la función f(x)=1/x, la cual se define como una hipérbola equilátera como se muestra
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Sin embargo, a diferencia de las polinomiales se puede observar que esta función no corta el eje de las “x”, es decir no hay ceros en la función. Al mismo tiempo puede verse que al hacerse la función de 0 el resultado es infinito: f(0)= ∞
Esto se podría comprobar al evaluar la función en valores cercanos a cero. Por ejemplo: F(x)=1/0.1=10
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Como la función no se puede evaluar en cero, el dominio de la función será: D= (-∞ ,+∞ ) – {0}. Así mismo el rango de la función será: R= (-∞,+∞) – {0}
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Función racional
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Una función racional es el cociente de dos funciones, f(x) y g(x) H(x)=f(x) y g(x) Donde f(x)y g(x) son funciones polinomiales y g(x) es diferente de cero. Esto significa que todos aquellos valores de “x” que hagan posible que g(x)=0 no será parte del dominio. Por eso se dice que el dominio de la función racional es el
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Así algunas funciones racionales serian las siguientes: F(x)=3/(x-2) G(x)=(5x-3)/(x2-4) H(x)=1+4/x + 3/x2
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EJEMPLO 1: Determina el dominio de la función: F(x)=3x/(x-2)
SOLUCION:
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EJEMPLO 2: Determina el dominio de la función: G(x)=(5x-3)/(x2-4)
SOLUCION:
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EJEMPLO 3: Determina el dominio de la función: H(x)=1+4/x + 3/x2 SOLUCION:
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En la figura 4 se puede observar que las ramas de la grafica se acercan hacia el valor en el que se encuentra el cero del denominador g(x), es decir a x=2 la rama de la derecha y a x=-1 y x=-3, la rama izquierda pero no toca en ese punto. En este caso se dice que esas rectas son las asíntotas verticales de la grafica. Al mismo tiempo se puede ver que las ramas se acercan hacia el eje X pero no lo tocan por eso se dice que la recta y=0 es la asíntota horizontal de la grafica
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Así a partir de los ejemplos anteriores y sus graficas se puede expr expresar esar lo siguiente: Una asíntota de una curva es una recta a la que la curva se acerca indefinidamente, a medida que X se acerca a un cierto valor sin llegar a tocarla Una asíntota vertical es una recta x=a de la grafica raciona
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Una asíntota horizontal es una recta y=b de la grafica de4 la función racional siempre que las ramas se acercan a b, a medida que X se acerca al +∞ o al -∞ Finalmente otro elemento que se puede determinar de las funciones racionales es determinar determ inar el rango ran go o el contradominio
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De la figura 1 se puede observar que los valores de G(x) varían desde -∞ hasta +∞, pero que no contienen el valor de y=0. por lo que el rango de F(x) son todos los reales
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De la figura 3 se puede observar que los valores de H(x) varían desde y=-1/3 aproximadamente apro ximadamente hasta +∞ por lo que el rango de H(x) es:(-1/3,+∞)
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Dada la función racional J(x)=8x+6)/(x2-x20) 1)Determina los ceros de la función g(x), el denominador
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SOLUCION: Los ceros de la función g(x)=(x2-x-20) Como x2-x-20 =(x-5)(x+4)= =(x-5)(x+4)=0 0 De donde se puede observar que en x=5 existe un
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El dominio de La función L(x) será D= R- (1,4)
El rango de la función es: [0,∞ )
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Ejemplo 4: Elaborar la grafica de la función M(x)=- √4-x, determina su dominio y su rango Solución: Se elabora la tabla de los valores similar a la siguiente x
-3
M(x) -√7
-2
-1
0
1
2
3
4
5
-√6
-√5
-√4
-√3
-√2
-√1
0
-√-1
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Grafica 4
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El dominio de la función será: (-∞ ,4] El rango de la función es: (-∞ ,0] Ejercicios De la grafica, el dominio y el rango de las siguientes funciones