Let's analyze the given functions:
[tex]\( f(x) = -3x^2 + x - 7 \)[/tex] and [tex]\( g(x) = 5x + 11 \)[/tex].
### Part (a)
To find [tex]\((f + g)(x)\)[/tex], we need to add the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (f + g)(x) = (-3x^2 + x - 7) + (5x + 11) \][/tex]
Combining like terms, we get:
[tex]\[ (f + g)(x) = -3x^2 + (x + 5x) + (-7 + 11) \][/tex]
[tex]\[ (f + g)(x) = -3x^2 + 6x + 4 \][/tex]
So, [tex]\((f + g)(x) = -3x^2 + 6x + 4\)[/tex].
### Part (b)
For the domain of [tex]\((f + g)(x)\)[/tex], we note that [tex]\((f + g)(x)\)[/tex] is a polynomial. The domain of any polynomial function is all real numbers.
Therefore, the domain of [tex]\((f + g)(x)\)[/tex] in interval notation is:
[tex]\[ \text{Domain of } (f + g)(x) = (-\infty, \infty) \][/tex]
### Part (c)
To find [tex]\((f - g)(x)\)[/tex], we need to subtract [tex]\( g(x) \)[/tex] from [tex]\( f(x) \)[/tex]:
[tex]\[ (f - g)(x) = (-3x^2 + x - 7) - (5x + 11) \][/tex]
Distributing the negative sign and combining like terms, we get:
[tex]\[ (f - g)(x) = -3x^2 + x - 7 - 5x - 11 \][/tex]
[tex]\[ (f - g)(x) = -3x^2 + (x - 5x) + (-7 - 11) \][/tex]
[tex]\[ (f - g)(x) = -3x^2 - 4x - 18 \][/tex]
So, [tex]\((f - g)(x) = -3x^2 - 4x - 18\)[/tex].
### Part (d)
Similar to [tex]\((f + g)(x)\)[/tex], [tex]\((f - g)(x)\)[/tex] is also a polynomial. The domain of any polynomial function is all real numbers.
Therefore, the domain of [tex]\((f - g)(x)\)[/tex] in interval notation is:
[tex]\[ \text{Domain of } (f - g)(x) = (-\infty, \infty) \][/tex]
### Part (e)
To find [tex]\((f \cdot g)(x)\)[/tex], we need to multiply [tex]\( f(x) \)[/tex] by [tex]\( g(x) \)[/tex]:
[tex]\[ (f \cdot g)(x) = (-3x^2 + x - 7) \cdot (5x + 11) \][/tex]
Expanding this product using the distributive property, we get:
[tex]\[ (f \cdot g)(x) = (-3x^2)(5x + 11) + (x)(5x + 11) + (-7)(5x + 11) \][/tex]
[tex]\[ (f \cdot g)(x) = -3x^2 \cdot 5x + (-3x^2 \cdot 11) + (x \cdot 5x) + (x \cdot 11) + (-7 \cdot 5x) + (-7 \cdot 11) \][/tex]
[tex]\[ (f \cdot g)(x) = -15x^3 - 33x^2 + 5x^2 + 11x - 35x - 77 \][/tex]
Combining like terms, we get:
[tex]\[ (f \cdot g)(x) = -15x^3 - 28x^2 - 24x - 77 \][/tex]
So, [tex]\((f \cdot g)(x) = (5x + 11)(-3x^2 + x - 7)\)[/tex].
### Part (f)
Again, [tex]\((f \cdot g)(x)\)[/tex] is a polynomial, and the domain of any polynomial function is all real numbers.
Therefore, the domain of [tex]\((f \cdot g)(x)\)[/tex] in interval notation is:
[tex]\[ \text{Domain of } (f \cdot g)(x) = (-\infty, \infty) \][/tex]
In summary:
a) [tex]\((f + g)(x) = -3x^2 + 6x + 4\)[/tex]
b) Domain of [tex]\((f + g)(x) = (-\infty, \infty)\)[/tex]
c) [tex]\((f - g)(x) = -3x^2 - 4x - 18\)[/tex]
d) Domain of [tex]\((f - g)(x) = (-\infty, \infty)\)[/tex]
e) [tex]\((f \cdot g)(x) = (5x + 11)(-3x^2 + x - 7)\)[/tex]
f) Domain of [tex]\((f \cdot g)(x) = (-\infty, \infty)\)[/tex]
