One-to-One Functions

According to our textbook, a function f is one-to-one if, for a and b in its domain,  f(a) = f(b)  implies that  a = b. 

To have an inverse, a function must be one-to-one. This means that no two elements in the domain of f correspond to the same element in the range of f. 


Testing for One-to-One Functions

Ex. Is g(x) = 3x - 2  one-to-one?

   See if g(a) = g(b) implies that a = b

   Substitute a for x and b for x. 

     6a - 2 = 6b - 2  (add 2 to both sides)

     6a = 6b   (Divide by 6)

       a = b

This is a one-to-one function!


Ex.  Is the function f(x) =   one-to-one?   

       (Multiply numerator by 5)

  (Simplify) 

   3a + 4 = 3b + 4   (Subtract 4 from both sides)


      3a = 3b     (Divide by 3)

     a = b

This is a one-to-one function!

Ex.  Is the function f(x) =  one-to-one?

     Let a and b be nonnegative real numbers with f(a) = f(b).

    

         (Subtract 1 from both sides)

        (Square both sides)

         a = b

Therefore,  f(a) = f(b), implies that a = b.   So,  f is one-to-one. 

Look at it this way:

 (0, 1)   (5, 2)   (6, 4)

Domain:  0, 5, 6

Range:  1, 2, 4

Each element in the domain (0, 5, and 6) correspond with a unique element in the range (1, 2, and 4)

Sometimes, always, never...

 An odd function is one-to-one... SOMETIMES

An even function is one-to-one... NEVER

Links to look at:

http://www.mathwarehouse.com/algebra/relation/one-to-one-function.php

http://www.mathwords.com/o/one_to_one_function.htm