1-to-1 Functions

**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?**

*y*^{2} = *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* = *x*^{2} +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* = 2*x* + 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*) = 2*x *+ 1, then

*f*(*p*) = *f*(*q*)

2(*p*) + 1 = 2(*q*) + 1

2(*p*) + 1 - 1 = 2(*q*) + 1 - 1

So, *p* = *q*.

**Let's see what happens when we try it with one of the other equations:**

If our function is *f*(*x*) = (2*x*^{2} + 1), then

*f*(*p*) = *f*(*q*)

(*p*^{2} + 1) = (*q*^{2} + 1)

(*p*^{2} + 1 - 1) = (*q*^{2} + 1 - 1)

*p*^{2} = *q*^{2}

The answer to a square root can either be positive or negative, so *p* and *q* are not necessarily equal.