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Learn the basic definition of a function and how to identify them using inputs and outputs.
Imagine a vending machine where you press the button for 'Cola' and sometimes you get a Cola, but other times you get a Lemon-Lime. Would you trust that machine? In mathematics, we only trust relationships that are 'predictable'—we call these functions.
In mathematics, a function is a special type of relationship between two sets of numbers. Think of it as a machine: you drop in an input (usually called ), the machine applies a specific rule, and it spits out an output (usually called ). The 'Golden Rule' that makes a relationship a function is this: Every input must have exactly one output. If you put the same number into the machine twice, you must get the same result both times. If an input ever points to two different values, the relationship is just a 'relation,' not a function.
Let's determine if the relationship between 'People' and 'Birthdays' is a function. 1. Let the input () be a person. 2. Let the output () be a birthday. 3. Since every person has exactly one birthday, this is a function. 4. However, if we reversed it (Input = Birthday, Output = Person), it would not be a function because one birthday can belong to many different people.
Quick Check
If the input produces an output , and later the same input produces an output , is this relationship a function?
Answer
No, because the single input 5 has two different outputs (10 and 12).
To identify a function in a table or a mapping diagram, you must act like a detective looking for 'cheating' inputs. Look at the column. If you see the same value appearing more than once, check its value. If the values are different, it's not a function! In a mapping diagram, this looks like one input having two or more arrows pointing away from it. Note: It is perfectly okay for two different inputs to share the same output (e.g., gives and also gives ).
Analyze this set of ordered pairs: . 1. List the inputs: . 2. Notice that the input appears twice. 3. Check the outputs for : They are and . 4. Because input has two different outputs, this is not a function.
Quick Check
In a mapping diagram, if two different arrows from two different inputs point to the same output, is it still a function?
Answer
Yes. Multiple inputs can share an output; only the reverse (one input to multiple outputs) is forbidden.
When we look at a relationship on a coordinate plane, we use a visual shortcut called the Vertical Line Test. Imagine sliding a vertical ruler across the graph from left to right. If at any point that vertical line touches the graph in more than one place, the graph is not a function. Why? Because a vertical line represents a single value. If it hits the graph twice, it means that value has two different values, which breaks our 'Golden Rule'.
Compare a circle and a diagonal line on a graph: 1. For a diagonal line (), any vertical line you draw will only cross the graph once. It is a function. 2. For a circle (), a vertical line through the center will hit the top and the bottom of the circle. Since it hits twice, a circle is not a function.
Which of the following sets of ordered pairs represents a function?
If a vertical line crosses a graph at points and , what can you conclude?
A mapping diagram where two arrows start at the same input and point to two different outputs represents a function.
Review Tomorrow
In 24 hours, try to explain the 'Vending Machine' analogy to a friend and draw a simple graph that fails the Vertical Line Test.
Practice Activity
Look at a collection of household items. Create a 'Function' where the Input is the item and the Output is its color. Does every item have exactly one color?