Horizontal Line Test
A function where each yvalue (output) corresponds to exactly one xvalue
(input) is called a onetoone function. A onetoone function has an
inverse function.
To determine if a function is onetoone, we can use the horizontal line
test.
Note:Another way to think about a function that
is onetoone is the following: Two
different input values always result in 2
different output values.
Procedure â€” To Determine if a Graph Represents a OneToOne Function
(Horizontal Line Test)
If you can draw a horizontal line anywhere through the graph of a
function and it intersects the graph at most once, then the graph
represents a onetoone function.
Example 1
Given the graph of the function: f(x) = 2x + 4
a. Is f(x) a onetoone function?
b. Does f(x) have an inverse?
Solution
a. To determine if the function is onetoone, use the horizontal line test.
Any horizontal line intersects the graph at most once. Thus, each
output corresponds to at most one input.
Therefore, f(x) = 2x + 4 is a onetoone function.
b. Since the function is onetoone, f(x) has an inverse.
Note:
Any linear function is onetoone unless it
has the form f(x) = c, where c is a
constant.
The graph of f(x) = c is a horizontal line
and so it does not pass the horizontal line
test.
Example 2
Given the graph of the function: f(x) = 1(x  3)^{2} + 4
a. Is f(x) a onetoone function?
b. Does f(x) have an inverse?
Solution
a. To determine if the function is onetoone, use the horizontal line test.
Since a horizontal line may intersect the graph more than once, some
outputs correspond to more than one input.
For example, the yvalue 3 corresponds to two xvalues (2 and 4).
Therefore, this function is not one to one.
b. Since the function is not onetoone, it does not have an inverse that is
a function.
