Pick three operations that can be performed on functions and explain the algebra and geometry behind each operation. Give an example of when to apply each operation.

### 1. **Function Addition**
   **Algebraic Explanation:**
   Function addition involves adding the outputs of two functions for each input in their domain. If \( f(x) \) and \( g(x) \) are two functions, then their sum \( h(x) \) is defined as:
   \[
   h(x) = f(x) + g(x)
   \]
   
   **Geometric Interpretation:**
   The graph of the resulting function \( h(x) \) is obtained by adding the y-values of \( f(x) \) and \( g(x) \) at each corresponding x-value. This can be visualized as shifting the graph of \( f(x) \) vertically by the y-values of \( g(x) \).

   **Example Application:**
   This operation is useful in modeling situations where two independent effects combine linearly. For example, if \( f(x) \) represents the cost function of materials in a project and \( g(x) \) represents the labor cost, then \( f(x) + g(x) \) would give the total cost function.

### 2. **Function Composition**
   **Algebraic Explanation:**
   Function composition involves applying one function to the result of another function. If \( f(x) \) and \( g(x) \) are two functions, the composition \( h(x) \) is defined as:
   \[
   h(x) = f(g(x))
   \]
   This means that the output of \( g(x) \) is used as the input for \( f(x) \).

   **Geometric Interpretation:**
   The graph of \( h(x) = f(g(x)) \) can be understood as a transformation of the graph of \( g(x) \) by \( f \). It is not as simple as a direct geometric translation or scaling; instead, it combines the geometric characteristics of both functions.

   **Example Application:**
   Function composition is useful when dealing with nested processes. For instance, if \( g(x) \) represents the temperature change over time and \( f(x) \) represents the rate of chemical reaction dependent on temperature, then \( f(g(x)) \) would describe how the reaction rate changes over time.

### 3. **Function Multiplication**
   **Algebraic Explanation:**
   Function multiplication involves multiplying the outputs of two functions for each input in their domain. If \( f(x) \) and \( g(x) \) are two functions, then their product \( h(x) \) is defined as:
   \[
   h(x) = f(x) \cdot g(x)
   \]

   **Geometric Interpretation:**
   The graph of \( h(x) \) is obtained by multiplying the y-values of \( f(x) \) and \( g(x) \) at each x-value. This can lead to a graph that stretches, shrinks, or even changes its shape significantly, depending on the values of \( f(x) \) and \( g(x) \).

   **Example Application:**
   Function multiplication is useful in scenarios involving proportional scaling. For example, if \( f(x) \) represents the revenue per product and \( g(x) \) represents the number of products sold, then \( f(x) \cdot g(x) \) gives the total revenue as a function of \( x \).

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### Do you want more details or have any questions? 

Here are five related questions to consider:

1. How do transformations like translation and scaling apply to function operations?
2. What is the difference between function addition and function subtraction geometrically?
3. Can you explain the difference between function composition and function multiplication in practical applications?
4. How does function division affect the domain and range compared to multiplication?
5. What are some common mistakes to avoid when performing function operations?

**Tip:** When performing operations on functions, always consider the domains of the individual functions to avoid undefined values in the result.

Explore three key operations on functions—addition, composition, and multiplication. Learn their algebraic explanations and geometric interpretations, and understand when to apply each operation. This page covers concepts suitable for Grades 10-12 students.

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