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The problem asks us to work with three given functions and perform operations on them. The functions provided are:

- \( f(x) = x + 5 \)
- \( g(x) = 2x - 1 \)
- \( h(x) = 2x^2 + 9x - 5 \)

We need to determine the following:

### a. \( (f + g)(x) \)
This represents the sum of the functions \( f(x) \) and \( g(x) \).

\[
(f + g)(x) = f(x) + g(x) = (x + 5) + (2x - 1) = 3x + 4
\]

### b. \( (f - g)(x) \)
This represents the difference of the functions \( f(x) \) and \( g(x) \).

\[
(f - g)(x) = f(x) - g(x) = (x + 5) - (2x - 1) = -x + 6
\]

### c. \( (f \cdot g)(x) \)
This represents the product of the functions \( f(x) \) and \( g(x) \).

\[
(f \cdot g)(x) = f(x) \times g(x) = (x + 5)(2x - 1)
\]
Expanding this:
\[
(f \cdot g)(x) = 2x^2 + 10x - x - 5 = 2x^2 + 9x - 5
\]

### d. \( \left(\frac{f}{g}\right)(x) \)
This represents the division of the functions \( f(x) \) and \( g(x) \).

\[
\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)} = \frac{x + 5}{2x - 1}
\]

### e. \( (f + g)(3) \)
Substitute \( x = 3 \) into \( (f + g)(x) = 3x + 4 \).

\[
(f + g)(3) = 3(3) + 4 = 9 + 4 = 13
\]

### f. \( (f - g)(3) \)
Substitute \( x = 3 \) into \( (f - g)(x) = -x + 6 \).

\[
(f - g)(3) = -(3) + 6 = -3 + 6 = 3
\]

### g. \( (f \cdot g)(3) \)
Substitute \( x = 3 \) into \( (f \cdot g)(x) = 2x^2 + 9x - 5 \).

\[
(f \cdot g)(3) = 2(3)^2 + 9(3) - 5 = 2(9) + 27 - 5 = 18 + 27 - 5 = 40
\]

### h. \( \left(\frac{f}{g}\right)(3) \)
Substitute \( x = 3 \) into \( \left(\frac{f}{g}\right)(x) = \frac{x + 5}{2x - 1} \).

\[
\left(\frac{f}{g}\right)(3) = \frac{3 + 5}{2(3) - 1} = \frac{8}{6 - 1} = \frac{8}{5}
\]

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Would you like further details on any of these calculations or additional practice questions? Here are 5 questions to expand your understanding:

1. What is the result of \( (f + h)(x) \)?
2. Find the derivative of \( (f \cdot g)(x) \).
3. Evaluate \( h(2) \).
4. Find the inverse of \( g(x) = 2x - 1 \).
5. Solve for \( x \) when \( (f - g)(x) = 0 \).

**Tip:** When performing operations on functions, remember to simplify each step as much as possible before substituting specific values.

Given the functions f(x) = x + 5, g(x) = 2x - 1, and h(x) = 2x^2 + 9x - 5, determine the following functions: (f + g)(x), (f - g)(x), (f * g)(x), (f / g)(x), (f + g)(3), (f - g)(3), (f * g)(3), (f / g)(3).

Learn how to perform basic operations on functions, including addition, subtraction, multiplication, and division. This example involves polynomial and rational functions, demonstrating the process with specific function values.

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