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### Given:
- \( f(x) = x^2 - 3x + 4 \)
- \( (f + g)(x) = 3x^2 + x - 5 \)

### To Find:
a. \( g(x) \)  
b. \( f(3) \) and \( g(3) \)  
c. \( (f + g)(3) \) and \( (f - g)(3) \)

---

### a. Finding \( g(x) \):

Since \( (f + g)(x) = f(x) + g(x) \), we can write:
\[
g(x) = (f + g)(x) - f(x)
\]

Substituting the given functions:
\[
g(x) = \left(3x^2 + x - 5\right) - \left(x^2 - 3x + 4\right)
\]
Simplifying the expression:
\[
g(x) = 3x^2 + x - 5 - x^2 + 3x - 4
\]
\[
g(x) = (3x^2 - x^2) + (x + 3x) + (-5 - 4)
\]
\[
g(x) = 2x^2 + 4x - 9
\]

So, \( g(x) = 2x^2 + 4x - 9 \).

### b. Finding \( f(3) \) and \( g(3) \):

- \( f(3) \):
\[
f(3) = 3^2 - 3(3) + 4 = 9 - 9 + 4 = 4
\]

- \( g(3) \):
\[
g(3) = 2(3^2) + 4(3) - 9 = 2(9) + 12 - 9 = 18 + 12 - 9 = 21
\]

So, \( f(3) = 4 \) and \( g(3) = 21 \).

### c. Finding \( (f + g)(3) \) and \( (f - g)(3) \):

- \( (f + g)(3) \):
\[
(f + g)(3) = f(3) + g(3) = 4 + 21 = 25
\]

- \( (f - g)(3) \):
\[
(f - g)(3) = f(3) - g(3) = 4 - 21 = -17
\]

So, \( (f + g)(3) = 25 \) and \( (f - g)(3) = -17 \).

---

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### Related Questions:
1. How do you find the difference of two functions?
2. What happens when you subtract a quadratic function from another quadratic function?
3. How does the value of \( g(x) \) change when you modify \( f(x) \) or \( (f + g)(x) \)?
4. Can you explain how to factor a quadratic function like \( g(x) \)?
5. What would happen if the coefficient of \( x^2 \) in \( (f + g)(x) \) were different?

**Tip:** Always substitute carefully when working with algebraic expressions, especially when subtracting functions.

Given f(x) = x^2 - 3x + 4 and (f + g)(x) = 3x^2 + x - 5, determine: a. g(x), b. f(3) and g(3), c. (f + g)(3) and (f - g)(3).

Learn how to solve algebraic problems involving function operations with quadratic equations. This detailed guide walks through finding g(x) from f(x) and (f + g)(x), calculating f(3) and g(3), and evaluating (f + g)(3) and (f - g)(3). Perfect for Grades 9-10 students.

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