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The image shows a math problem involving operations on functions. You are asked to perform the following operations on two functions:

- \( f(x) = 2x + 3 \)
- \( g(x) = x^2 - 1 \)

Here are the requested operations:

### a. \( (f + g)(x) \)
This represents the sum of the two functions:
\[
(f + g)(x) = f(x) + g(x) = (2x + 3) + (x^2 - 1)
\]
Simplifying:
\[
(f + g)(x) = x^2 + 2x + 2
\]

### b. \( (f - g)(x) \)
This represents the difference between the two functions:
\[
(f - g)(x) = f(x) - g(x) = (2x + 3) - (x^2 - 1)
\]
Simplifying:
\[
(f - g)(x) = -x^2 + 2x + 4
\]

### c. \( (fg)(x) \)
This represents the product of the two functions:
\[
(fg)(x) = f(x) \cdot g(x) = (2x + 3)(x^2 - 1)
\]
Using the distributive property (FOIL):
\[
(fg)(x) = 2x(x^2) + 2x(-1) + 3(x^2) + 3(-1) = 2x^3 - 2x + 3x^2 - 3
\]
Simplifying:
\[
(fg)(x) = 2x^3 + 3x^2 - 2x - 3
\]

### d. \( \left( \frac{f}{g} \right)(x) \)
This represents the quotient of the two functions:
\[
\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)} = \frac{2x + 3}{x^2 - 1}
\]
No further simplification is possible.

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Would you like more details on any of these operations or have any questions?

### Related Questions:
1. What is the domain of \( (f + g)(x) \)?
2. How can we find the inverse of \( f(x) \)?
3. How do you solve \( f(g(x)) \), the composition of these two functions?
4. What happens when \( g(x) = 0 \) in the quotient \( \frac{f}{g}(x) \)?
5. Can we graph these functions to visualize their behavior?

### Tip:
When working with operations on functions, always consider the domain, especially when dividing functions or dealing with square roots or logarithms.

Let f(x) = 2x + 3 and g(x) = x^2 - 1. Find the following operations: (f + g)(x), (f - g)(x), (fg)(x), and (f/g)(x).

Learn how to perform operations on the functions f(x) = 2x + 3 and g(x) = x^2 - 1. This problem covers addition, subtraction, multiplication, and division of functions, suitable for high school students in Grades 10-12.

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