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The image presents a question involving two functions \( f(x) = 2x + 3 \) and \( g(x) = 7x + 1 \), asking to find different combinations of these functions. Let's solve each part:

### A. \( (f + g)(x) \)
This represents the sum of the functions:
\[
(f + g)(x) = f(x) + g(x) = (2x + 3) + (7x + 1) = 9x + 4
\]

### B. \( (f - g)(x) \)
This represents the difference between the functions:
\[
(f - g)(x) = f(x) - g(x) = (2x + 3) - (7x + 1) = -5x + 2
\]

### C. \( (fg)(x) \)
This represents the product of the functions:
\[
(fg)(x) = f(x) \times g(x) = (2x + 3) \times (7x + 1) 
\]
Expanding this:
\[
fg(x) = 14x^2 + 2x + 21x + 3 = 14x^2 + 23x + 3
\]

### D. \( \left(\frac{f}{g}\right)(x) \)
This represents the quotient of the functions:
\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{2x + 3}{7x + 1}
\]

### E. \( f \circ g(x) \)
This represents the composition of the functions \( f(g(x)) \):
\[
f(g(x)) = f(7x + 1) 
\]
Substituting \( g(x) \) into \( f(x) \):
\[
f(7x + 1) = 2(7x + 1) + 3 = 14x + 2 + 3 = 14x + 5
\]

Thus, the answers to the given options are:
- **A**: \( 9x + 4 \)
- **B**: \( -5x + 2 \)
- **C**: \( 14x^2 + 23x + 3 \)
- **D**: \( \frac{2x + 3}{7x + 1} \)
- **E**: \( 14x + 5 \)

Would you like any more details or explanations?

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**Related Questions:**

1. How do you compute the composition of functions \( g \circ f(x) \)?
2. What happens if you subtract \( g(x) \) from \( f(x) \) when the functions are polynomials of higher degrees?
3. How does the product of two quadratic functions look like, in general?
4. What is the domain of the quotient \( \frac{f(x)}{g(x)} \) for these functions?
5. How does function composition affect the domain and range of a function?

**Tip:** When dealing with function operations, always simplify the expressions where possible, and carefully consider the domains of the functions, especially in compositions and divisions.

If f(x) = 2x + 3 and g(x) = 7x + 1, find the following: A. (f + g)(x), B. (f - g)(x), C. (fg)(x), D. (f/g)(x), E. f ∘ g(x).

Learn how to solve function operations and composition for the functions f(x) = 2x + 3 and g(x) = 7x + 1. This problem covers key algebraic concepts including addition, subtraction, multiplication, division, and composition of functions. Suitable for students in Grades 9-11.

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