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We are given two functions \( f(x) = \sqrt{x+5} \) and \( g(x) = (x+2)(x-3) \), and we need to compute the following:

### a. \( (f+g)(x) \)
This is the sum of the two functions:
\[
(f+g)(x) = f(x) + g(x)
\]
Substitute the values of \( f(x) \) and \( g(x) \):
\[
(f+g)(x) = \sqrt{x+5} + (x+2)(x-3)
\]
Expand \( g(x) \):
\[
g(x) = (x+2)(x-3) = x^2 - 3x + 2x - 6 = x^2 - x - 6
\]
Thus,
\[
(f+g)(x) = \sqrt{x+5} + x^2 - x - 6
\]

### b. \( (f-g)(x) \)
This is the difference of the two functions:
\[
(f-g)(x) = f(x) - g(x)
\]
Substitute the values of \( f(x) \) and \( g(x) \):
\[
(f-g)(x) = \sqrt{x+5} - (x^2 - x - 6)
\]
Simplify:
\[
(f-g)(x) = \sqrt{x+5} - x^2 + x + 6
\]

### c. \( (f \cdot g)(x) \)
This is the product of the two functions:
\[
(f \cdot g)(x) = f(x) \cdot g(x)
\]
Substitute the values of \( f(x) \) and \( g(x) \):
\[
(f \cdot g)(x) = \sqrt{x+5} \cdot (x^2 - x - 6)
\]

### d. \( \left( \frac{f}{g} \right)(x) \)
This is the quotient of the two functions:
\[
\left( \frac{f}{g} \right)(x) = \frac{f(x)}{g(x)}
\]
Substitute the values of \( f(x) \) and \( g(x) \):
\[
\left( \frac{f}{g} \right)(x) = \frac{\sqrt{x+5}}{x^2 - x - 6}
\]

Would you like me to provide further details or evaluate any specific part?

Here are 5 related questions to consider:
1. What is the domain of the function \( f(x) = \sqrt{x+5} \)?
2. How does expanding \( g(x) = (x+2)(x-3) \) affect the combined functions?
3. What is the significance of function composition in algebraic functions?
4. How can we find the points of intersection between \( f(x) \) and \( g(x) \)?
5. What methods are used to simplify or solve the quotient \( \frac{f(x)}{g(x)} \)?

**Tip:** Always check the domain restrictions for combined functions, especially when dealing with square roots and denominators.

f(x) = sqrt(x + 5) and g(x) = (x + 2)(x - 3), find: a. (f + g)(x), b. (f - g)(x), c. (f * g)(x), d. (f / g)(x)

Learn how to find the sum, difference, product, and quotient of two functions, f(x) = sqrt(x + 5) and g(x) = (x + 2)(x - 3). This solution involves algebraic concepts such as quadratic expansion and function operations. Suitable for students in Grades 10-12.

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