Everyday usage and use in mathematics
In everyday language, the word function often refers to a specific feature or capability of a device or tool. For example, a smartphone might have a camera function that allows it to take photos; or, we could say the primary function of a toaster is to make toast. This use of function describes a particular task or role that the device can perform. Similarly, a mathematical function describes a particular set of tasks that modify an input. For example, consider the mathematical function \[ f(x) = x^2 \] \[ f(3) = 9 \]
In this case, the function takes an input \( x \) and performs the task of squaring it, producing the output \( x^2 \). If we give this function a real number, for example, \( 3 \), it will return an output, \( 9 \). In the next example, we imagine a toaster as a mathmatical function. \[ \text{toaster}(x) = x + \text{heat} \] In this example, we could say \( x = \text{bread} \), and our function would return \( \text{toast} \). \[ \text{toaster}(bread) = \text{toast} \]
The word function is translated into Spanish as función. The everyday meaning of función in Spanish is similar to the English meaning in both its literal and figurative usage.
Summary
In both everyday and mathematical contexts, a function 'does' something specific and predictable, performing a defined role. In mathematics, a function carries out a set of operations on an input.
Usage in Mathematics
More specifically, the term function refers to the relationship between two variables: all possible inputs, and their outputs. A function includes an equation that describes how each input is related to its output; one side of the equation usually declares a variable (most common '\(y\) ' or '\(f(x)\)'), and the other side describes the changes that make the two variables equal (such as \(+2\) or \( \sqrt x \)).
Functions allow us to model real-world phenomena and understand how changes in one quantity affect another. A function can take various forms, such as linear, quadratic, or exponential, each describing a different type of relationship.
Examples
For instance, the linear function \[ f(x) = 2x + 3 \] represents a straight-line relationship between \( x \) and \( f(x) \). The quadratic function \[ g(x) = x^2 - 4x + 4 \] represents a parabolic relationship. Each of these functions performs a specific role in mapping inputs to outputs.
Exercises
- Describe the mathematical and everyday meaning of the word function in your native language. Try to explain how a mathematical function is related to the everyday definition of function as a specific task or role.
- What might the output of this function be? \( \text{blender}(banana + milk) = x \)
- Write a mathematical function that would result in these inputs and outputs.
Input Output 2 4 Sock Pair Singer Duo Semester Year
References
[1] Oxford English Dictionary, s.v. “function, n.”,https://www.oed.com/dictionary/function_n