S(x) = x+3  and t(x) =2x+6 write the expressions for (s-t)(x) and (s•t)(x) and evaluate (s+t)(1)

Given the functions \( S(x) = x + 3 \) and \( t(x) = 2x + 6 \), we need to find the expressions for \((S - t)(x)\) and \((S \cdot t)(x)\) and evaluate \((S + t)(1)\).

### Step-by-Step Solutions:

1. **Expression for \((S - t)(x)\):**
   \[
   (S - t)(x) = S(x) - t(x)
   \]
   Substitute the given functions \( S(x) \) and \( t(x) \):
   \[
   (S - t)(x) = (x + 3) - (2x + 6)
   \]
   Simplify the expression:
   \[
   (S - t)(x) = x + 3 - 2x - 6 = -x - 3
   \]

2. **Expression for \((S \cdot t)(x)\):**
   \[
   (S \cdot t)(x) = S(x) \cdot t(x)
   \]
   Substitute the given functions \( S(x) \) and \( t(x) \):
   \[
   (S \cdot t)(x) = (x + 3) \cdot (2x + 6)
   \]
   Use the distributive property to expand the expression:
   \[
   (S \cdot t)(x) = x \cdot 2x + x \cdot 6 + 3 \cdot 2x + 3 \cdot 6
   \]
   Simplify the expression:
   \[
   (S \cdot t)(x) = 2x^2 + 6x + 6x + 18 = 2x^2 + 12x + 18
   \]

3. **Evaluate \((S + t)(1)\):**
   \[
   (S + t)(x) = S(x) + t(x)
   \]
   Substitute the given functions \( S(x) \) and \( t(x) \):
   \[
   (S + t)(x) = (x + 3) + (2x + 6)
   \]
   Simplify the expression:
   \[
   (S + t)(x) = x + 3 + 2x + 6 = 3x + 9
   \]
   Now, evaluate this at \( x = 1 \):
   \[
   (S + t)(1) = 3(1) + 9 = 3 + 9 = 12
   \]

### Summary of Results:
1. \((S - t)(x) = -x - 3\)
2. \((S \cdot t)(x) = 2x^2 + 12x + 18\)
3. \((S + t)(1) = 12\)

Would you like more details or have any questions about these solutions?

### Related Questions:
1. How do you find the composition of two functions, such as \( S(t(x)) \)?
2. What is the inverse of the function \( S(x) \)?
3. How do you graph the functions \( S(x) \) and \( t(x) \)?
4. Can you find the derivative of the function \( S(x) \)?
5. How do you solve for \( x \) when \( S(x) = t(x) \)?
6. What are the intercepts of the function \( t(x) \)?
7. How do you determine the domain and range of the function \( S(x) \)?
8. What is the result of adding the functions \( S(x) \) and \( t(x) \)?

### Tip:
When dealing with functions, always carefully substitute and simplify expressions step-by-step to avoid errors.

Explore arithmetic operations on functions S(x) = x + 3 and t(x) = 2x + 6. Learn how to add, subtract, and multiply these functions, with step-by-step solutions and evaluation of (S + t)(1). Suitable for students in Grades 9-12.

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