Functions

What is a Function?

• A function is a rule that assigns to each input value exactly one output value.

• Think of it like a machine: you put something in (input), and it produces a specific result (output).

Key Concepts

• Input: The value that is given to the function. Often represented by the variable 'x'.

• Output: The value that the function produces. Often represented by 'f(x)' (read as "f of x").

• Domain: The set of all possible input values.

• Range: The set of all possible output values.

• Function Notation:

  • Typically written as: f(x) = ...

  • Example: f(x) = 2x + 1

    • This means that the function 'f' takes an input 'x', multiplies it by 2, and then adds 1.

Types of Functions

• Linear Functions:

  • Represented by equations of the form: f(x) = mx + b

  • Graph is a straight line.

• Quadratic Functions:

  • Represented by equations of the form: f(x) = ax² + bx + c

  • Graph is a parabola.

• Exponential Functions:

  • Involve exponents, such as: f(x) = 2^x

  • Used to model growth and decay.

• Trigonometric Functions:

  • Deal with angles and sides of triangles (sine, cosine, tangent, etc.).

Applications of Functions

• Modeling real-world phenomena:

  • Population growth, physical laws, financial models.

• Computer science:

  • Programming, algorithms.

• Engineering:

  • Designing systems, analyzing data.