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Sagot :
Claro, procedamos a determinar los elementos del conjunto [tex]\(M\)[/tex].
Paso 1: Definir el rango de [tex]\(x\)[/tex]
El rango de [tex]\(x\)[/tex] se define entre [tex]\(A-2\)[/tex] y 9, donde [tex]\(A\)[/tex] es un valor dado. Supongamos que [tex]\(A = 2\)[/tex].
Por lo tanto, el rango de [tex]\(x\)[/tex] es: [tex]\(A-2 \leq x \leq 9\)[/tex]
[tex]\[2 - 2 \leq x \leq 9\][/tex]
[tex]\[0 \leq x \leq 9\][/tex]
Entonces, [tex]\(x\)[/tex] toma los valores: 0, 1, 2, 3, 4, 5, 6, 7, 8, y 9.
Paso 2: Aplicar la expresión [tex]\(\left( \frac{x^2 - 1}{2} \right)\)[/tex] para cada valor de [tex]\(x\)[/tex]
Vamos a evaluar la expresión para cada [tex]\(x\)[/tex]:
1. Para [tex]\(x = 0\)[/tex]:
[tex]\[\left( \frac{0^2 - 1}{2} \right) = \left( \frac{-1}{2} \right) = -1\][/tex]
2. Para [tex]\(x = 1\)[/tex]:
[tex]\[\left( \frac{1^2 - 1}{2} \right) = \left( \frac{0}{2} \right) = 0\][/tex]
3. Para [tex]\(x = 2\)[/tex]:
[tex]\[\left( \frac{2^2 - 1}{2} \right) = \left( \frac{4 - 1}{2} \right) = \left( \frac{3}{2} \right) = 1\][/tex]
4. Para [tex]\(x = 3\)[/tex]:
[tex]\[\left( \frac{3^2 - 1}{2} \right) = \left( \frac{9 - 1}{2} \right) = \left( \frac{8}{2} \right) = 4\][/tex]
5. Para [tex]\(x = 4\)[/tex]:
[tex]\[\left( \frac{4^2 - 1}{2} \right) = \left( \frac{16 - 1}{2} \right) = \left( \frac{15}{2} \right) = 7\][/tex]
6. Para [tex]\(x = 5\)[/tex]:
[tex]\[\left( \frac{5^2 - 1}{2} \right) = \left( \frac{25 - 1}{2} \right) = \left( \frac{24}{2} \right) = 12\][/tex]
7. Para [tex]\(x = 6\)[/tex]:
[tex]\[\left( \frac{6^2 - 1}{2} \right) = \left( \frac{36 - 1}{2} \right) = \left( \frac{35}{2} \right) = 17\][/tex]
8. Para [tex]\(x = 7\)[/tex]:
[tex]\[\left( \frac{7^2 - 1}{2} \right) = \left( \frac{49 - 1}{2} \right) = \left( \frac{48}{2} \right) = 24\][/tex]
9. Para [tex]\(x = 8\)[/tex]:
[tex]\[\left( \frac{8^2 - 1}{2} \right) = \left( \frac{64 - 1}{2} \right) = \left( \frac{63}{2} \right) = 31\][/tex]
10. Para [tex]\(x = 9\)[/tex]:
[tex]\[\left( \frac{9^2 - 1}{2} \right) = \left( \frac{81 - 1}{2} \right) = \left( \frac{80}{2} \right) = 40\][/tex]
Conclusión:
Los elementos del conjunto [tex]\(M\)[/tex] son los siguientes:
[tex]\[ M = \{-1, 0, 1, 4, 7, 12, 17, 24, 31, 40\} \][/tex]
Así, hemos determinado los elementos del conjunto [tex]\(M\)[/tex] de manera detallada y paso a paso.
Paso 1: Definir el rango de [tex]\(x\)[/tex]
El rango de [tex]\(x\)[/tex] se define entre [tex]\(A-2\)[/tex] y 9, donde [tex]\(A\)[/tex] es un valor dado. Supongamos que [tex]\(A = 2\)[/tex].
Por lo tanto, el rango de [tex]\(x\)[/tex] es: [tex]\(A-2 \leq x \leq 9\)[/tex]
[tex]\[2 - 2 \leq x \leq 9\][/tex]
[tex]\[0 \leq x \leq 9\][/tex]
Entonces, [tex]\(x\)[/tex] toma los valores: 0, 1, 2, 3, 4, 5, 6, 7, 8, y 9.
Paso 2: Aplicar la expresión [tex]\(\left( \frac{x^2 - 1}{2} \right)\)[/tex] para cada valor de [tex]\(x\)[/tex]
Vamos a evaluar la expresión para cada [tex]\(x\)[/tex]:
1. Para [tex]\(x = 0\)[/tex]:
[tex]\[\left( \frac{0^2 - 1}{2} \right) = \left( \frac{-1}{2} \right) = -1\][/tex]
2. Para [tex]\(x = 1\)[/tex]:
[tex]\[\left( \frac{1^2 - 1}{2} \right) = \left( \frac{0}{2} \right) = 0\][/tex]
3. Para [tex]\(x = 2\)[/tex]:
[tex]\[\left( \frac{2^2 - 1}{2} \right) = \left( \frac{4 - 1}{2} \right) = \left( \frac{3}{2} \right) = 1\][/tex]
4. Para [tex]\(x = 3\)[/tex]:
[tex]\[\left( \frac{3^2 - 1}{2} \right) = \left( \frac{9 - 1}{2} \right) = \left( \frac{8}{2} \right) = 4\][/tex]
5. Para [tex]\(x = 4\)[/tex]:
[tex]\[\left( \frac{4^2 - 1}{2} \right) = \left( \frac{16 - 1}{2} \right) = \left( \frac{15}{2} \right) = 7\][/tex]
6. Para [tex]\(x = 5\)[/tex]:
[tex]\[\left( \frac{5^2 - 1}{2} \right) = \left( \frac{25 - 1}{2} \right) = \left( \frac{24}{2} \right) = 12\][/tex]
7. Para [tex]\(x = 6\)[/tex]:
[tex]\[\left( \frac{6^2 - 1}{2} \right) = \left( \frac{36 - 1}{2} \right) = \left( \frac{35}{2} \right) = 17\][/tex]
8. Para [tex]\(x = 7\)[/tex]:
[tex]\[\left( \frac{7^2 - 1}{2} \right) = \left( \frac{49 - 1}{2} \right) = \left( \frac{48}{2} \right) = 24\][/tex]
9. Para [tex]\(x = 8\)[/tex]:
[tex]\[\left( \frac{8^2 - 1}{2} \right) = \left( \frac{64 - 1}{2} \right) = \left( \frac{63}{2} \right) = 31\][/tex]
10. Para [tex]\(x = 9\)[/tex]:
[tex]\[\left( \frac{9^2 - 1}{2} \right) = \left( \frac{81 - 1}{2} \right) = \left( \frac{80}{2} \right) = 40\][/tex]
Conclusión:
Los elementos del conjunto [tex]\(M\)[/tex] son los siguientes:
[tex]\[ M = \{-1, 0, 1, 4, 7, 12, 17, 24, 31, 40\} \][/tex]
Así, hemos determinado los elementos del conjunto [tex]\(M\)[/tex] de manera detallada y paso a paso.
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