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.