Give a problem and solution for this Operation of function

Sure! Let's work with the concept of **operation on functions**, where two functions are combined using arithmetic operations like addition, subtraction, multiplication, or division. Here's an example problem and its solution:

### Problem:
Given two functions:
\[
f(x) = 2x + 3
\]
\[
g(x) = x^2 - 4
\]
Find the function \( h(x) \) when:
1. \( h(x) = f(x) + g(x) \)
2. \( h(x) = f(x) - g(x) \)
3. \( h(x) = f(x) \times g(x) \)
4. \( h(x) = \frac{f(x)}{g(x)} \) (for \( g(x) \neq 0 \))

### Solution:

1. **Addition**:
   \[
   h(x) = f(x) + g(x)
   \]
   Substitute \( f(x) \) and \( g(x) \):
   \[
   h(x) = (2x + 3) + (x^2 - 4) = x^2 + 2x - 1
   \]

2. **Subtraction**:
   \[
   h(x) = f(x) - g(x)
   \]
   Substitute \( f(x) \) and \( g(x) \):
   \[
   h(x) = (2x + 3) - (x^2 - 4) = -x^2 + 2x + 7
   \]

3. **Multiplication**:
   \[
   h(x) = f(x) \times g(x)
   \]
   Substitute \( f(x) \) and \( g(x) \):
   \[
   h(x) = (2x + 3) \times (x^2 - 4)
   \]
   Expand the product:
   \[
   h(x) = 2x(x^2 - 4) + 3(x^2 - 4) = 2x^3 - 8x + 3x^2 - 12 = 2x^3 + 3x^2 - 8x - 12
   \]

4. **Division**:
   \[
   h(x) = \frac{f(x)}{g(x)}
   \]
   Substitute \( f(x) \) and \( g(x) \):
   \[
   h(x) = \frac{2x + 3}{x^2 - 4}
   \]
   Since \( g(x) = x^2 - 4 = (x - 2)(x + 2) \), \( h(x) \) is defined for \( x \neq 2 \) and \( x \neq -2 \).

### Do you want any details or have any questions?  
Here are five related questions for you to explore:

1. What happens to the function \( h(x) \) when \( f(x) = g(x) \)?
2. How would you find the zeros of the function \( h(x) \) for any of the operations above?
3. What is the domain of the function \( h(x) = \frac{f(x)}{g(x)} \) when \( f(x) = x + 1 \) and \( g(x) = x^2 - 1 \)?
4. How do these operations on functions apply to composite functions?
5. Can you graph the functions \( f(x) \), \( g(x) \), and \( h(x) \) to visualize their behavior?

**Tip:** When performing operations on functions, always consider the domain of the resulting function, especially in division where the denominator cannot be zero.

Explore the operation of functions problem and solution, including addition, subtraction, multiplication, and division of functions. Learn how to combine functions like f(x) = 2x + 3 and g(x) = x^2 - 4 to find h(x) for various operations. Suitable for Grades 10-12.

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