One to one functions

In algebra 1 and algebra 2, the topic of one to one functions often comes up. We know that if a function is one to one, it has an inverse, which can be useful knowledge.

Graphical approaches:

Use horizontal and vertical line tests. Draw a vertical line through the graph and it should only intersect with function at one point. Draw a horizontal line through the graph and it should only intersect at one point.

example of a function that is not one to one (fails horizontal line test):

x square

example of a one to one function (passes both horizontal and vertical line tests):

3^x

Algebraic approach:

Algebraically, how to test for one to one:

 

Take g(x)= 1/x for example:

1.it’s one to one if g(a)=g(b)

  1. plug in: g(a)=1/a and g(b)=1/b.
  2. set them equal to each other: 1/a=1/b
  3. if you multiply both sides by a, and both sides by b, you get a=b.
  4. Thus, it’s one to one.
  5. And we can confirm it graphically:
1/X

Function above : 1/X