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4. Halla [tex]\( f(-3) \)[/tex]; [tex]\( f(0) \)[/tex]; [tex]\( f(m) \)[/tex]; [tex]\( f(x+1) \)[/tex]; [tex]\( f(\Delta x) \)[/tex]; [tex]\( f\left(x^2\right) \)[/tex]; [tex]\( f(x+\Delta x) \)[/tex] en cada una de las siguientes funciones:

a. [tex]\( f(x) = x^2 \)[/tex]

b. [tex]\( f(x) = x^2 - 2 \)[/tex]

c. [tex]\( f(x) = x^2 + x + 1 \)[/tex]

d. [tex]\( f(x) = x^2 \)[/tex]

e. [tex]\( f(x) = -(x+1)^2 \)[/tex]

f. [tex]\( f(x) = \frac{x^2}{2} \)[/tex]



Answer :

GinnyAnswer
Vamos a evaluar las diferentes funciones dadas para los valores especificados. Procederemos paso a paso para cada función [tex]\(f(x)\)[/tex].

### a) [tex]\(f(x) = x^2\)[/tex]

1. [tex]\(f(-3) = (-3)^2 = 9\)[/tex]
2. [tex]\(f(0) = 0^2 = 0\)[/tex]
3. [tex]\(f(m) = m^2\)[/tex]
4. [tex]\(f(x+1) = (x+1)^2 = x^2 + 2x + 1\)[/tex]
5. [tex]\(f(\Delta x) = (\Delta x)^2 = \Delta x^2\)[/tex]
6. [tex]\(f(x^2) = (x^2)^2 = x^4\)[/tex]
7. [tex]\(f(x+\Delta x) = (x + \Delta x)^2 = x^2 + 2x\Delta x + (\Delta x)^2\)[/tex]

### b) [tex]\(f(x) = x^2 - 2\)[/tex]

1. [tex]\(f(-3) = (-3)^2 - 2 = 9 - 2 = 7\)[/tex]
2. [tex]\(f(0) = 0^2 - 2 = 0 - 2 = -2\)[/tex]
3. [tex]\(f(m) = m^2 - 2\)[/tex]
4. [tex]\(f(x+1) = (x+1)^2 - 2 = x^2 + 2x + 1 - 2 = x^2 + 2x - 1\)[/tex]
5. [tex]\(f(\Delta x) = (\Delta x)^2 - 2 = \Delta x^2 - 2\)[/tex]
6. [tex]\(f(x^2) = (x^2)^2 - 2 = x^4 - 2\)[/tex]
7. [tex]\(f(x+\Delta x) = (x + \Delta x)^2 - 2 = x^2 + 2x\Delta x + (\Delta x)^2 - 2\)[/tex]

### c) [tex]\(f(x) = x^2 + x + 1\)[/tex]

1. [tex]\(f(-3) = (-3)^2 + (-3) + 1 = 9 - 3 + 1 = 7\)[/tex]
2. [tex]\(f(0) = 0^2 + 0 + 1 = 1\)[/tex]
3. [tex]\(f(m) = m^2 + m + 1\)[/tex]
4. [tex]\(f(x+1) = (x+1)^2 + (x+1) + 1 = x^2 + 2x + 1 + x + 1 + 1 = x^2 + 3x + 3\)[/tex]
5. [tex]\(f(\Delta x) = (\Delta x)^2 + \Delta x + 1 = \Delta x^2 + \Delta x + 1\)[/tex]
6. [tex]\(f(x^2) = (x^2)^2 + x^2 + 1 = x^4 + x^2 + 1\)[/tex]
7. [tex]\(f(x+\Delta x) = (x + \Delta x)^2 + (x + \Delta x) + 1 = x^2 + 2x\Delta x + (\Delta x)^2 + x + \Delta x + 1\)[/tex]

### d) [tex]\(f(x) = x^2\)[/tex]

El mismo que el caso a) porque es la misma función:

1. [tex]\(f(-3) = (-3)^2 = 9\)[/tex]
2. [tex]\(f(0) = 0^2 = 0\)[/tex]
3. [tex]\(f(m) = m^2\)[/tex]
4. [tex]\(f(x+1) = (x+1)^2 = x^2 + 2x + 1\)[/tex]
5. [tex]\(f(\Delta x) = (\Delta x)^2 = \Delta x^2\)[/tex]
6. [tex]\(f(x^2) = (x^2)^2 = x^4\)[/tex]
7. [tex]\(f(x+\Delta x) = (x + \Delta x)^2 = x^2 + 2x\Delta x + (\Delta x)^2\)[/tex]

### e) [tex]\(f(x) = -(x+1)^2\)[/tex]

1. [tex]\(f(-3) = -((-3)+1)^2 = -( -2)^2 = -4\)[/tex]
2. [tex]\(f(0) = -(0+1)^2 = -(1)^2 = -1\)[/tex]
3. [tex]\(f(m) = -(m+1)^2\)[/tex]
4. [tex]\(f(x+1) = -((x+1)+1)^2 = -(x+2)^2\)[/tex]
5. [tex]\(f(\Delta x) = -((\Delta x)+1)^2 = -(\Delta x + 1)^2\)[/tex]
6. [tex]\(f(x^2) = -(x^2 + 1)^2\)[/tex]
7. [tex]\(f(x+\Delta x) = -((x+\Delta x)+1)^2 = -(x + \Delta x + 1)^2\)[/tex]

### f) [tex]\(f(x) = \frac{x^2}{2}\)[/tex]

1. [tex]\(f(-3) = \frac{(-3)^2}{2} = \frac{9}{2}\)[/tex]
2. [tex]\(f(0) = \frac{0^2}{2} = 0\)[/tex]
3. [tex]\(f(m) = \frac{m^2}{2}\)[/tex]
4. [tex]\(f(x+1) = \frac{(x+1)^2}{2} = \frac{x^2 + 2x + 1}{2} = \frac{x^2}{2} + x + \frac{1}{2}\)[/tex]
5. [tex]\(f(\Delta x) = \frac{(\Delta x)^2}{2} = \frac{\Delta x^2}{2}\)[/tex]
6. [tex]\(f(x^2) = \frac{(x^2)^2}{2} = \frac{x^4}{2}\)[/tex]
7. [tex]\(f(x+\Delta x) = \frac{(x+\Delta x)^2}{2} = \frac{x^2 + 2x\Delta x + (\Delta x)^2}{2} = \frac{x^2}{2} + x\Delta x + \frac{\Delta x^2}{2}\)[/tex]

Estas son las evaluaciones de las funciones dadas en los puntos específicos.

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