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.

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.