1-to-1 Functions

A 1-to-1 function passes a vertical line test and a horizontal line test.

This record is not due to good test-taking skills, but it is due to the function's domain and range. Every x-value of a 1:1 function can yield only one possible value of y (as with any function), and every y-value corresponds with only one value of x.

Which of the following graphs shows a 1:1 function?

 

y² = x

This equation does not pass the vertical line test, so it is not a function. On this graph, most values of x have more than one possible value of y.

 

 

 

y = x² +1

This equation is a function, but it does not pass the horizontal line test. On this graph, most values of y have more than one possible value of x.

 

 

 

y = 2x + 1

The line above represents a 1-to-1 function because each value of x can only yield a single value of y, and vice versa.

 

But what if you don't have a graphing calculator? How can you determine if a function is a 1-to-1 function?!

 

Mathematicians have come up with a test:

If f(p) = f(q), then p = q.  If this fact holds, then congratulations! You have a 1-to-1 function.

If our function is f(x) = 2x + 1, then

f(p) = f(q)

2p + 1 = 2q + 1

 2p + 1 − 1 = 2q + 1 − 1

 2p = 2q

p = q

The fact holds, so the function is 1-to-1.

Let's try another example:

 

If our function is f(x) = 2x² + 1, then

f(p) = f(q)

p² + 1 = q² + 1

p² + 1 − 1 = q² + 1 − 1

p² = q²

±p = ±q

The answer to a square root can either be positive or negative, so p and q are not necessarily equal.  Either p = q, or p = −q.  Therefore, the fact does not hold, and the function is not 1-to-1.