Vocabulary
Function
Domain
Range
Domain
Range
Functions, Domain, and Range
A fundamental concept covered in math courses is understanding functions. A function is a mathematical operation that takes an input, does something to it, and gives you an output.
You could imagine this as a machine in a factory; something goes down a conveyor belt into the machine, and a *thud* *bang* *whir* *hiss* later, a new thing comes out.
You could imagine this as a machine in a factory; something goes down a conveyor belt into the machine, and a *thud* *bang* *whir* *hiss* later, a new thing comes out.
A mathematical function takes a number, variable, or object and "does something to it." Examples of functions could be that it multiplies a number by 2, or adds 5, or a combination of different things.
The domain of a function is the mathematical word for the inputs. The domain can be called different names: inputs, x-values, or the independent variable.
The range of a function is the mathematical word for the outputs. The range can be called different names: outputs, y-values, or the dependent variable.
Functions are described two ways: we give it a name and we describe what the function does. When we name a function, we like to name the function according to what the function's domain and range is. For instance, we could name a function that relates the amount of gas we put in our car to how much that would cost. So, we could say the cost of gas depends on the amount of gas I put in the car.
In general, we name functions by abbreviating output(input). This generally looks like f(x), where the f is abbreviated for function. You can also see "y" instead of "f(x)", because y is the output of the function called f(x).
When we describe what a function does, that is the mathematical operation. Let's stick with the gas example. If gas costs $1.99 per gallon, then we would say that the function takes our input (gas) and multiplies it by $1.99, or $1.99g.
Using this whole scenario, we would call the function C(g)=$1.99g. The variable g is the amount of gas you put in, and the output of C(g) is how much you spend.
The images below summarizes the concepts of functions and their domain and range.
The domain of a function is the mathematical word for the inputs. The domain can be called different names: inputs, x-values, or the independent variable.
The range of a function is the mathematical word for the outputs. The range can be called different names: outputs, y-values, or the dependent variable.
Functions are described two ways: we give it a name and we describe what the function does. When we name a function, we like to name the function according to what the function's domain and range is. For instance, we could name a function that relates the amount of gas we put in our car to how much that would cost. So, we could say the cost of gas depends on the amount of gas I put in the car.
- Therefore, mathematically, we would say: The cost of gas is a function of the amount of gas I put in the car.
- However, whenever we're doing math, we can abbreviate that thought. We can abbreviate that to Cost(gas).
- We can abbreviate it even further: C(g).
In general, we name functions by abbreviating output(input). This generally looks like f(x), where the f is abbreviated for function. You can also see "y" instead of "f(x)", because y is the output of the function called f(x).
When we describe what a function does, that is the mathematical operation. Let's stick with the gas example. If gas costs $1.99 per gallon, then we would say that the function takes our input (gas) and multiplies it by $1.99, or $1.99g.
Using this whole scenario, we would call the function C(g)=$1.99g. The variable g is the amount of gas you put in, and the output of C(g) is how much you spend.
The images below summarizes the concepts of functions and their domain and range.
Identifying and Representing Functions
There are three key characteristics that a function must have in order to be a function:
There are different ways to represent a function. For this lesson, we will explore three different representations: tables, mapping diagrams, and rules.
- A domain - functions have to have numbers to put into it.
- A range - functions should have numbers that come out.
- Every input has exactly one output - if I put a number in, I should always get the same number out.
There are different ways to represent a function. For this lesson, we will explore three different representations: tables, mapping diagrams, and rules.
Tables
A table shows an input and its corresponding output.
Look at the slideshow of tables to see different examples of how possible functions are represented. |
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Mapping Diagrams
A mapping diagram shows a collection of inputs and outputs, with arrows from the inputs pointing to their output.
Look at the slideshow of mapping diagrams to see different examples of how possible functions are represented. |
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Function Rules |
Function rules are basically the equation that relates the input and output. Below are different exampl
Summary Questions
1. What is the domain of a function? What is the range?
2. What are the three key features of a function?
2. What are the three key features of a function?
More Resources
Missouri Learning Standards
A1.IF.A.1a - Understand that a function from one set (domain) to another set (range) assigns to each element of the domain exactly one element of the range, a) represent a function using function notation.