In mathematics, a function is generally written as an equation showing the relationship between an input and an output.
Understanding Functions
A function describes a relationship where each input value corresponds to exactly one output value. Think of it as a machine: you put something in (the input), and the machine gives you something back (the output).
- Domain: The set of all possible input values.
- Codomain/Range: The set of all possible output values.
General Notation
The most common way to represent a function is using the notation f(x).
- f: Represents the name of the function.
- x: Represents the input variable.
- f(x): Represents the output of the function for the given input x. According to the reference, this is related to the output y as in: y = f(x).
Components of a Function
Here's a breakdown of the key components with examples:
| Component | Description | Example |
|---|---|---|
| Function Name | A letter or symbol representing the function. | f, g, h |
| Input Variable | The variable that is fed into the function. | x, t, n |
| Function Output | The result of applying the function to the input variable. | f(x), g(t), h(n) |
| Equation | The rule that defines how the input relates to the output. | f(x) = x2 + 2x + 1 |
Examples
Here are some examples illustrating how functions are written:
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Example 1: A simple linear function.
f(x) = 2x + 3- This function multiplies the input x by 2 and then adds 3. If x is 2, then f(2) = 2(2) + 3 = 7.
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Example 2: A quadratic function.
g(x) = x^2 - 4x + 4- This function squares the input x, subtracts 4 times x, and then adds 4.
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Example 3: A trigonometric function.
h(x) = sin(x)- This function returns the sine of the input x.
How to Write a Function: Steps
- Choose a Function Name: Select a letter (usually f, g, or h) to represent the function.
- Define the Input Variable: Identify the input variable (usually x).
- Write the Equation: Express the relationship between the input and output using a mathematical equation. This equation shows what operations are performed on the input to produce the output.
- Express the Function: Combine the function name, input variable, and equation to define the function: f(x) = [equation].
