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We are given two functions:

\[
s(x) = \frac{x - 2}{x^2 - 25} \quad \text{and} \quad t(x) = \frac{x - 5}{2 - x}
\]

We need to:

1. Find \( (s + t)(x) \), which is the sum of the two functions.
2. Write the domain of the sum in interval notation.

### Step 1: Finding \( (s + t)(x) \)

We add the two functions:

\[
(s + t)(x) = s(x) + t(x) = \frac{x - 2}{x^2 - 25} + \frac{x - 5}{2 - x}
\]

#### Simplify the denominator:

First, factor \( x^2 - 25 \):

\[
x^2 - 25 = (x - 5)(x + 5)
\]

Now the sum looks like:

\[
(s + t)(x) = \frac{x - 2}{(x - 5)(x + 5)} + \frac{x - 5}{2 - x}
\]

Notice that \( 2 - x = -(x - 2) \). So we rewrite the second term:

\[
\frac{x - 5}{2 - x} = \frac{x - 5}{-(x - 2)} = -1
\]

Thus, the sum simplifies to:

\[
(s + t)(x) = \frac{x - 2}{(x - 5)(x + 5)} - 1
\]

To combine these terms, express \( -1 \) as a fraction:

\[
-1 = \frac{-(x - 5)(x + 5)}{(x - 5)(x + 5)}
\]

Now add the fractions:

\[
(s + t)(x) = \frac{x - 2 - (x - 5)(x + 5)}{(x - 5)(x + 5)}
\]

Simplify the numerator:

\[
x - 2 - (x^2 - 25) = x - 2 - x^2 + 25 = -x^2 + x + 23
\]

Thus, the final expression for \( (s + t)(x) \) is:

\[
(s + t)(x) = \frac{-x^2 + x + 23}{(x - 5)(x + 5)}
\]

### Step 2: Find the Domain

To determine the domain, we look for the values of \( x \) that make the denominator zero:

- \( x - 5 = 0 \) gives \( x = 5 \),
- \( x + 5 = 0 \) gives \( x = -5 \),
- \( 2 - x = 0 \) gives \( x = 2 \).

Therefore, the function is undefined at \( x = 5 \), \( x = -5 \), and \( x = 2 \).

The domain of \( (s + t)(x) \) is all real numbers except \( x = 5 \), \( x = -5 \), and \( x = 2 \).

In interval notation, the domain is:

\[
(-\infty, -5) \cup (-5, 2) \cup (2, 5) \cup (5, \infty)
\]

Would you like more details on any part? Here are 5 questions you might want to explore next:

1. How can you verify the domain of a function based on its equation?
2. How do you simplify complex rational expressions like the one in this problem?
3. What does it mean for a function to be undefined at certain points?
4. How do you find the sum of two functions with different denominators?
5. How does factoring help simplify algebraic expressions?

**Tip:** Always check for common factors in the numerator and denominator before combining fractions to simplify the process!

Refer to functions s and t. Find the indicated function and write the domain in interval notation. Write your answer as a single fraction.

Discover how to add rational functions s(x) = (x - 2)/(x^2 - 25) and t(x) = (x - 5)/(2 - x) and find their combined domain. Learn the steps to simplify their sum and identify values excluded from the domain, suitable for high school students in Grades 10-12.

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