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We are given the functions:

\[
s(x) = 3x + 2
\]
\[
t(x) = x - 3
\]

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

\[
(s + t)(x) = s(x) + t(x)
\]
Substitute the expressions for \( s(x) \) and \( t(x) \):

\[
(s + t)(x) = (3x + 2) + (x - 3)
\]
\[
(s + t)(x) = 3x + x + 2 - 3 = 4x - 1
\]

### 2. Finding \( (s \cdot t)(x) \):

\[
(s \cdot t)(x) = s(x) \cdot t(x)
\]
Substitute the expressions for \( s(x) \) and \( t(x) \):

\[
(s \cdot t)(x) = (3x + 2)(x - 3)
\]
Apply the distributive property (FOIL):

\[
(s \cdot t)(x) = 3x(x - 3) + 2(x - 3)
\]
\[
(s \cdot t)(x) = 3x^2 - 9x + 2x - 6
\]
\[
(s \cdot t)(x) = 3x^2 - 7x - 6
\]

### 3. Evaluating \( (s - t)(-1) \):

\[
(s - t)(x) = s(x) - t(x)
\]
Substitute the expressions for \( s(x) \) and \( t(x) \):

\[
(s - t)(x) = (3x + 2) - (x - 3)
\]
Distribute the negative sign:

\[
(s - t)(x) = 3x + 2 - x + 3 = 2x + 5
\]
Now, substitute \( x = -1 \):

\[
(s - t)(-1) = 2(-1) + 5 = -2 + 5 = 3
\]

### Final Answers:
\[
(s + t)(x) = 4x - 1
\]
\[
(s \cdot t)(x) = 3x^2 - 7x - 6
\]
\[
(s - t)(-1) = 3
\]

Would you like more details on any of these steps or have further questions?

Here are 5 related questions for practice:
1. What is \( (s - t)(x) \) in its general form?
2. How would you find \( \frac{s}{t}(x) \)?
3. Can you find the derivative of \( s(x) \) and \( t(x) \)?
4. What is the value of \( (s + t)(2) \)?
5. How would you graph the function \( s(x) - t(x) \)?

**Tip:** Always simplify the terms after adding or multiplying functions to ensure the expression is in its simplest form.

Suppose that the functions s and t are defined for all real numbers x as follows: s(x) = 3x + 2, t(x) = x - 3. Write the expressions for (s + t)(x) and (s · t)(x) and evaluate (s - t)(-1).

Learn how to perform algebraic operations on functions with a detailed solution. The problem involves finding the expressions for (s + t)(x) and (s · t)(x), and evaluating (s - t)(-1), using s(x) = 3x + 2 and t(x) = x - 3. Suitable for students in Grades 8-10.

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