Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Get immediate and reliable answers to your questions from a community of experienced experts on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
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]
[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]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
