Find (f+g)(x) and (f-g)(x) using two functions given in each number.                2. f(x)=5x+1 g(x)=-2+3x

To find \((f+g)(x)\) and \((f-g)(x)\) using the functions \(f(x) = 5x + 1\) and \(g(x) = -2 + 3x\), follow these steps:

### 1. Find \((f+g)(x)\):
\[
(f+g)(x) = f(x) + g(x)
\]
Substitute the given functions:
\[
(f+g)(x) = (5x + 1) + (-2 + 3x)
\]
Combine like terms:
\[
(f+g)(x) = 5x + 3x + 1 - 2 = 8x - 1
\]
So,
\[
(f+g)(x) = 8x - 1
\]

### 2. Find \((f-g)(x)\):
\[
(f-g)(x) = f(x) - g(x)
\]
Substitute the given functions:
\[
(f-g)(x) = (5x + 1) - (-2 + 3x)
\]
Distribute the negative sign:
\[
(f-g)(x) = 5x + 1 + 2 - 3x
\]
Combine like terms:
\[
(f-g)(x) = 5x - 3x + 1 + 2 = 2x + 3
\]
So,
\[
(f-g)(x) = 2x + 3
\]

### Summary:
- \((f+g)(x) = 8x - 1\)
- \((f-g)(x) = 2x + 3\)

Would you like more details or have any questions?

Here are five related questions to further explore this topic:
1. How would you find \((f \times g)(x)\) using the same functions?
2. What is the graph of \((f+g)(x)\) and \((f-g)(x)\)?
3. How do you find the domain and range of \((f+g)(x)\) and \((f-g)(x)\)?
4. What happens if you divide \(f(x)\) by \(g(x)\) and find \(\frac{f}{g}(x)\)?
5. Can you solve for the values of \(x\) where \((f+g)(x) = 0\) and \((f-g)(x) = 0\)?

**Tip:** When subtracting one function from another, carefully distribute the negative sign across all terms to avoid mistakes.

Learn how to find (f+g)(x) and (f-g)(x) using the functions f(x) = 5x + 1 and g(x) = -2 + 3x. Detailed step-by-step solution with combining like terms and distributing negative signs. Suitable for students in Grades 9-12 studying algebra and operations on functions.

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