A1.F-IF.A Understand the concept of a function and use function notation.

### Understanding the Concept of a Function and Using Function Notation (A1.F-IF.A)

#### 1. **Definition of a Function**  
A **function** is a relation between a set of inputs (domain) and a set of possible outputs (range) such that **each input corresponds to exactly one output**. In other words, for every $ x $ in the domain, there is a unique $ y $ in the range.

#### 2. **Function Notation**  
Functions are typically written as $ f(x) $, where:
- $ f $ is the name of the function.
- $ x $ is the input (independent variable).
- $ f(x) $ is the output (dependent variable), read as "the value of $ f $ at $ x $."

For example, if $ f(x) = 2x + 3 $, then:
- When $ x = 2 $, $ f(2) = 2(2) + 3 = 7 $.
- $ f(2) = 7 $ means the function $ f $ assigns $ 7 $ to the input $ 2 $.

#### 3. **Key Properties of Functions**  
- **Domain**: The set of all possible input values ($ x $) for which the function is defined.  
- **Range**: The set of all possible output values ($ f(x) $).  
- **Vertical Line Test**: A graph represents a function if no vertical line intersects the graph more than once.  

#### 4. **Examples of Functions**  
- **Linear Function**: $ f(x) = mx + b $ (e.g., $ f(x) = 3x + 1 $)  
- **Quadratic Function**: $ f(x) = ax^2 + bx + c $ (e.g., $ f(x) = x^2 - 4 $)  
- **Non-Function**: $ x^2 + y^2 = 1 $ is not a function because a single $ x $-value corresponds to two $ y $-values.

#### 5. **Using Function Notation in Real Life**  
Functions are used to model real-world situations. For instance:
- If $ C(t) $ represents the cost (\$) as a function of time $ t $ (hours), $ C(2) $ tells us the cost after 2 hours.

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### Questions for Deeper Understanding:
1. How can you identify the domain and range of a function from its equation or graph?  
2. What is the significance of the vertical line test in determining if a graph represents a function?  
3. How does function notation simplify representing relationships in mathematical problems?  
4. Can you provide examples of real-world scenarios where functions are used?  
5. What happens when a relation is not a function? How can you recognize it?  

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### Tip:  
Always remember that in a function, **every input has one and only one output**, but different inputs can share the same output.

Explore the concept of functions and function notation, including key properties like domain, range, and the vertical line test. This guide explains linear and quadratic functions with examples and real-world applications. Ideal for Grades 8-11.

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