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?

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.