3.5
Rules for Defining Functions
A relation is a function if each input x is associated with exactly one output y. For example, the equation
y = 2x + 5 defines a function because ever…
An equation defines a function when each input from the domain gives only one output in the range. If a single input from the domain gives two different outputs from the range, the relation is not a function.
Function tables follow the same rule, if a table gives more than one output for the same input, it does not define a function.
On a graph, if two points share the same x-value but have different y-values, then it is not a function.
A graph represents a function if it satisfies the vertical line test, meaning no vertical line intersects the graph more than once.
For example, a sideways parabola fails the vertical line test because vertical lines intersect it at two points.
Splitting this parabola into upper and lower parts creates two separate functions. Each one passes the vertical line test.
Reversing the roles of x and y in a sideways parabola can define x as a function of y as long as each y-value corresponds to only one x-value.
For example, a working vending machine is a great illustration of a function: each button pressed from the given domain corresponds to one and only one specific snack from the range.
Q1: What makes a relation a function?
A relation is a function when each input from the domain corresponds to exactly one output in the range. If a single input produces two different outputs, the relation fails to be a function. This one-to-one correspondence is the fundamental requirement for any equation, table, or graph to define a function.
Q2: How does the vertical line test determine if a graph is a function?
The vertical line test checks whether any vertical line intersects a graph more than once. If a vertical line crosses the curve at two or more points, the graph fails the test and does not represent a function. A sideways parabola, for example, fails because vertical lines intersect it at two different points.
Q3: Can a non-function be converted into separate functions?
Yes, a non-function can often be split into multiple valid functions. For instance, a sideways parabola that fails the vertical line test can be divided into upper and lower parts, each passing the test independently. Similarly, the equation y² = x can be split into two separate functions.
Q4: Why doesn't a function table with repeated inputs define a function?
A function table must assign exactly one output to each input. If the same input appears multiple times with different outputs, the table violates the definition of a function. This principle applies whether data is presented in a table, equation, or graph format.
Q5: What is an example of a function in everyday life?
A vending machine illustrates function behavior perfectly. Each button pressed from the available domain corresponds to exactly one specific snack from the range. This one-to-one relationship between input and output mirrors how mathematical functions operate and relates to introduction to one to one functions.
Q6: How can reversing x and y values create a function from a non-function?
Reversing the roles of x and y can define x as a function of y instead. For a sideways parabola, treating y as the input and x as the output allows each y-value to correspond to only one x-value, satisfying the function definition and passing the vertical line test.
Q7: What distinguishes an equation that defines a function from one that does not?
An equation defines a function if every x-value produces exactly one y-value. For example, y = 2x + 5 is a function because each input yields a unique output. However, x = y² + 1 is not a function of x because a single x-value corresponds to multiple y-values.