Determine the sum and difference of each functions.                            f(x)=\frac{x+5}{x-3}

g(x)=\frac{x+4}{x-2}

To determine the sum and difference of the functions \( f(x) = \frac{x+5}{x-3} \) and \( g(x) = \frac{x+4}{x-2} \), we proceed as follows:

### Sum of the Functions
The sum of the functions \( f(x) \) and \( g(x) \) is given by:

\[
f(x) + g(x) = \frac{x+5}{x-3} + \frac{x+4}{x-2}
\]

To add these fractions, we need to find a common denominator, which is \( (x-3)(x-2) \).

\[
f(x) + g(x) = \frac{(x+5)(x-2)}{(x-3)(x-2)} + \frac{(x+4)(x-3)}{(x-3)(x-2)}
\]

Now, combine the numerators:

\[
f(x) + g(x) = \frac{(x+5)(x-2) + (x+4)(x-3)}{(x-3)(x-2)}
\]

Next, expand the numerators:

\[
(x+5)(x-2) = x^2 - 2x + 5x - 10 = x^2 + 3x - 10
\]

\[
(x+4)(x-3) = x^2 - 3x + 4x - 12 = x^2 + x - 12
\]

Now, add the expanded numerators:

\[
f(x) + g(x) = \frac{(x^2 + 3x - 10) + (x^2 + x - 12)}{(x-3)(x-2)} = \frac{2x^2 + 4x - 22}{(x-3)(x-2)}
\]

This can be simplified further:

\[
f(x) + g(x) = \frac{2(x^2 + 2x - 11)}{(x-3)(x-2)}
\]

### Difference of the Functions
The difference of the functions \( f(x) \) and \( g(x) \) is given by:

\[
f(x) - g(x) = \frac{x+5}{x-3} - \frac{x+4}{x-2}
\]

Again, find a common denominator:

\[
f(x) - g(x) = \frac{(x+5)(x-2)}{(x-3)(x-2)} - \frac{(x+4)(x-3)}{(x-3)(x-2)}
\]

Now, combine the numerators:

\[
f(x) - g(x) = \frac{(x+5)(x-2) - (x+4)(x-3)}{(x-3)(x-2)}
\]

Using the previously expanded forms:

\[
f(x) - g(x) = \frac{(x^2 + 3x - 10) - (x^2 + x - 12)}{(x-3)(x-2)} = \frac{(x^2 + 3x - 10) - (x^2 + x - 12)}{(x-3)(x-2)}
\]

Simplify the numerator:

\[
f(x) - g(x) = \frac{x^2 + 3x - 10 - x^2 - x + 12}{(x-3)(x-2)} = \frac{2x + 2}{(x-3)(x-2)}
\]

This can be simplified further:

\[
f(x) - g(x) = \frac{2(x + 1)}{(x-3)(x-2)}
\]

### Final Answers:
- **Sum of the functions**: \( f(x) + g(x) = \frac{2(x^2 + 2x - 11)}{(x-3)(x-2)} \)
- **Difference of the functions**: \( f(x) - g(x) = \frac{2(x + 1)}{(x-3)(x-2)} \)

Would you like further details or explanations on any part of this? 

Here are some related questions for further exploration:
1. What is the product of \( f(x) \) and \( g(x) \)?
2. How do you determine the quotient of \( f(x) \) and \( g(x) \)?
3. Can you find the domain of \( f(x) + g(x) \)?
4. What is the domain of \( f(x) - g(x) \)?
5. How would the graphs of \( f(x) \) and \( g(x) \) look like?

**Tip**: Always simplify your expressions as much as possible to make solving and interpreting them easier.

Explore how to find the sum and difference of functions f(x) = (x+5)/(x-3) and g(x) = (x+4)/(x-2) with detailed step-by-step solutions. Learn about adding and subtracting fractions with unlike denominators and simplifying algebraic expressions. Suitable for high school students and anyone interested in algebraic function operations.

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