Question
Find f plus g, f minus g, fg and StartFraction f Over g EndFraction
. Determine the domain for each function.
f left parenthesis x right parenthesis equals 3 x minus 7, g left parenthesis x right parenthesis equals x minus 3
Question content area bottom
Part 1
left parenthesis f plus g right parenthesis left parenthesis x right parenthesis equals
enter your response here (Simplify your answer.)
Part 2
What is the domain of f plus g?
A.
The domain of f plus g is StartSet nothing EndSet
. (Use a comma to separate answers as needed.)
B.
The domain of f plus g is
enter your response here. (Type your answer in interval notation.)
C.
The domain of f plus g is empty set.
Part 3
left parenthesis f minus g right parenthesis left parenthesis x right parenthesis equals
enter your response here (Simplify your answer.)
Part 4
What is the domain of f minus g?
A.
The domain of f minus g is
enter your response here. (Type your answer in interval notation.)
B.
The domain of f minus g is StartSet nothing EndSet
. (Use a comma to separate answers as needed.)
C.
The domain of f minus g is empty set.
Part 5
left parenthesis fg right parenthesis left parenthesis x right parenthesis equals
enter your response here
(Simplify your answer.)
Part 6
What is the domain of fg?
A.
The domain of fg is
enter your response here. (Type your answer in interval notation.)
B.
The domain of fg is StartSet nothing EndSet
. (Use a comma to separate answers as needed.)
C.
The domain of fg is empty set.
Part 7
left parenthesis StartFraction f Over g EndFraction right parenthesis left parenthesis x right parenthesis equals
enter your response here (Simplify your answer.)
Part 8
What is the domain of StartFraction f Over g EndFraction
?
A.
The domain of StartFraction f Over g EndFraction
is
enter your response here. (Type your answer in interval notation.)
B.
The domain of StartFraction f Over g EndFraction
is StartSet nothing EndSet
. (Use a comma to separate answers as needed.)
C.
The domain of StartFraction f Over g EndFraction
is empty set.
. Determine the domain for each function.
f left parenthesis x right parenthesis equals 3 x minus 7, g left parenthesis x right parenthesis equals x minus 3
Question content area bottom
Part 1
left parenthesis f plus g right parenthesis left parenthesis x right parenthesis equals
enter your response here (Simplify your answer.)
Part 2
What is the domain of f plus g?
A.
The domain of f plus g is StartSet nothing EndSet
. (Use a comma to separate answers as needed.)
B.
The domain of f plus g is
enter your response here. (Type your answer in interval notation.)
C.
The domain of f plus g is empty set.
Part 3
left parenthesis f minus g right parenthesis left parenthesis x right parenthesis equals
enter your response here (Simplify your answer.)
Part 4
What is the domain of f minus g?
A.
The domain of f minus g is
enter your response here. (Type your answer in interval notation.)
B.
The domain of f minus g is StartSet nothing EndSet
. (Use a comma to separate answers as needed.)
C.
The domain of f minus g is empty set.
Part 5
left parenthesis fg right parenthesis left parenthesis x right parenthesis equals
enter your response here
(Simplify your answer.)
Part 6
What is the domain of fg?
A.
The domain of fg is
enter your response here. (Type your answer in interval notation.)
B.
The domain of fg is StartSet nothing EndSet
. (Use a comma to separate answers as needed.)
C.
The domain of fg is empty set.
Part 7
left parenthesis StartFraction f Over g EndFraction right parenthesis left parenthesis x right parenthesis equals
enter your response here (Simplify your answer.)
Part 8
What is the domain of StartFraction f Over g EndFraction
?
A.
The domain of StartFraction f Over g EndFraction
is
enter your response here. (Type your answer in interval notation.)
B.
The domain of StartFraction f Over g EndFraction
is StartSet nothing EndSet
. (Use a comma to separate answers as needed.)
C.
The domain of StartFraction f Over g EndFraction
is empty set.
Answers
GPT-4o mini
Let's solve each part step by step.
### Given:
- \( f(x) = 3x - 7 \)
- \( g(x) = x - 3 \)
### Part 1: \( (f + g)(x) \)
To find \( f + g \), we add the two functions:
\[
(f + g)(x) = f(x) + g(x) = (3x - 7) + (x - 3) = 3x - 7 + x - 3 = 4x - 10
\]
So,
\[
(f + g)(x) = 4x - 10
\]
### Part 2: Domain of \( f + g \)
The domain of \( f(x) \) and \( g(x) \) is all real numbers, so:
\[
\text{Domain of } (f + g) = (-\infty, \infty)
\]
### Part 3: \( (f - g)(x) \)
To find \( f - g \), we subtract \( g \) from \( f \):
\[
(f - g)(x) = f(x) - g(x) = (3x - 7) - (x - 3) = 3x - 7 - x + 3 = 2x - 4
\]
So,
\[
(f - g)(x) = 2x - 4
\]
### Part 4: Domain of \( f - g \)
Again, the domain of both functions is all real numbers:
\[
\text{Domain of } (f - g) = (-\infty, \infty)
\]
### Part 5: \( (fg)(x) \)
To find \( fg \), we multiply \( f \) and \( g \):
\[
(fg)(x) = f(x) \cdot g(x) = (3x - 7)(x - 3)
\]
Now expanding this:
\[
= 3x^2 - 9x - 7x + 21 = 3x^2 - 16x + 21
\]
So,
\[
(fg)(x) = 3x^2 - 16x + 21
\]
### Part 6: Domain of \( fg \)
The domain for the product of two polynomials is also all real numbers:
\[
\text{Domain of } fg = (-\infty, \infty)
\]
### Part 7: \( \left(\frac{f}{g}\right)(x) \)
For the quotient, we find \( \frac{f}{g} \):
\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{3x - 7}{x - 3}
\]
### Part 8: Domain of \( \frac{f}{g} \)
The function \( \frac{f}{g} \) is defined wherever \( g(x) \neq 0 \):
\[
g(x) = x - 3 = 0 \Rightarrow x = 3 \quad \text{is a point of discontinuity.}
\]
So, the domain in interval notation is:
\[
\text{Domain of } \frac{f}{g} = (-\infty, 3) \cup (3, \infty)
\]
### Summary of Answers
1. \( (f + g)(x) = 4x - 10 \)
2. Domain of \( f + g \): \((- \infty, \infty)\)
3. \( (f - g)(x) = 2x - 4 \)
4. Domain of \( f - g \): \((- \infty, \infty)\)
5. \( (fg)(x) = 3x^2 - 16x + 21 \)
6. Domain of \( fg \): \((- \infty, \infty)\)
7. \( \left(\frac{f}{g}\right)(x) = \frac{3x - 7}{x - 3} \)
8. Domain of \( \frac{f}{g} \): \((- \infty, 3) \cup (3, \infty)\)
### Given:
- \( f(x) = 3x - 7 \)
- \( g(x) = x - 3 \)
### Part 1: \( (f + g)(x) \)
To find \( f + g \), we add the two functions:
\[
(f + g)(x) = f(x) + g(x) = (3x - 7) + (x - 3) = 3x - 7 + x - 3 = 4x - 10
\]
So,
\[
(f + g)(x) = 4x - 10
\]
### Part 2: Domain of \( f + g \)
The domain of \( f(x) \) and \( g(x) \) is all real numbers, so:
\[
\text{Domain of } (f + g) = (-\infty, \infty)
\]
### Part 3: \( (f - g)(x) \)
To find \( f - g \), we subtract \( g \) from \( f \):
\[
(f - g)(x) = f(x) - g(x) = (3x - 7) - (x - 3) = 3x - 7 - x + 3 = 2x - 4
\]
So,
\[
(f - g)(x) = 2x - 4
\]
### Part 4: Domain of \( f - g \)
Again, the domain of both functions is all real numbers:
\[
\text{Domain of } (f - g) = (-\infty, \infty)
\]
### Part 5: \( (fg)(x) \)
To find \( fg \), we multiply \( f \) and \( g \):
\[
(fg)(x) = f(x) \cdot g(x) = (3x - 7)(x - 3)
\]
Now expanding this:
\[
= 3x^2 - 9x - 7x + 21 = 3x^2 - 16x + 21
\]
So,
\[
(fg)(x) = 3x^2 - 16x + 21
\]
### Part 6: Domain of \( fg \)
The domain for the product of two polynomials is also all real numbers:
\[
\text{Domain of } fg = (-\infty, \infty)
\]
### Part 7: \( \left(\frac{f}{g}\right)(x) \)
For the quotient, we find \( \frac{f}{g} \):
\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{3x - 7}{x - 3}
\]
### Part 8: Domain of \( \frac{f}{g} \)
The function \( \frac{f}{g} \) is defined wherever \( g(x) \neq 0 \):
\[
g(x) = x - 3 = 0 \Rightarrow x = 3 \quad \text{is a point of discontinuity.}
\]
So, the domain in interval notation is:
\[
\text{Domain of } \frac{f}{g} = (-\infty, 3) \cup (3, \infty)
\]
### Summary of Answers
1. \( (f + g)(x) = 4x - 10 \)
2. Domain of \( f + g \): \((- \infty, \infty)\)
3. \( (f - g)(x) = 2x - 4 \)
4. Domain of \( f - g \): \((- \infty, \infty)\)
5. \( (fg)(x) = 3x^2 - 16x + 21 \)
6. Domain of \( fg \): \((- \infty, \infty)\)
7. \( \left(\frac{f}{g}\right)(x) = \frac{3x - 7}{x - 3} \)
8. Domain of \( \frac{f}{g} \): \((- \infty, 3) \cup (3, \infty)\)