A function is a mathematical concept that describes a relation between a set of inputs and a set of possible outputs, such that each input is related to exactly one output. In simpler terms, a function assigns each element from one set (the domain) to exactly one element in another set (the codomain).
There are different types of functions, and here are some common types:
1. Linear Function:
- Definition: A function that can be represented by a linear equation, typically in the form \(f(x) = mx + b\), where \(m\) and \(b\) are constants.
- Example: \(f(x) = 2x + 3\) is a linear function.
2. Quadratic Function:
- Definition: A function that can be represented by a quadratic equation, typically in the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.
- Example: \(f(x) = x^2 - 4\) is a quadratic function.
3. Exponential Function:
- Definition: A function where the variable is an exponent. It is often written as \(f(x) = a^x\), where \(a\) is a constant.
- Example: \(f(x) = 2^x\) is an exponential function.
4. Trigonometric Function:
- Definition: Functions involving trigonometric ratios (sine, cosine, tangent, etc.) of an angle.
- Example: \(f(x) = \sin(x)\) is a trigonometric function.
5. Piecewise Function:
- Definition: A function that is defined by different formulas for different ranges of the input.
- Example:
\[
f(x) =
\begin{cases}
x, & \text{if } x < 0 \\
x^2, & \text{if } x \geq 0
\end{cases}
\]
These are just a few examples, and there are many other types of functions, each serving different mathematical purposes.
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