Explore the concept of one-to-one function using examples. This concept is
necessary to understand the concept of inverse function definition and its
properties and also to solve certain types of
equations. Several functions are explored graphically using the horizontal line test. Examples of analytical explanations are, in some
cases, provided to support the graphical approach followed here. Some definitions are reviewed so that the explorations can be carried out without difficulties.
## One to One Function## Definition of a FunctionA function is a rule that produces a correspondence between the elements of two sets: D ( domain ) and R ( range ), such that to each element in D there corresponds one and only one element in R.
## Definition of a one-to-one functionA function is a one-to-one if no two different elements in D have the same element in R.The definition of a one to one function can be written algebraically as follows: Let x _{1} and x_{2} be any elements of D
A function f(x) is one-to-one
if x _{1} is not equal to x_{2} then f(x_{1}) is not equal to f(x_{2})
Using the contrapositive to the above
A function f(x) is one-to-one
This last property is useful in proving that a function is or is not a one to one.
if f(x _{1}) = f(x_{2}) then x_{1} = x_{2}.
In the Venn diagram below, function f is a one to one since not two inputs have a common output. In the Venn diagram below, function f is NOT a one to one since the inputs -1 and 0 have the same output. ## Horizontal Line TestIf the graph of a function is known, it is fairly easy to determine if that function is a one to one or not using the horizontal line test.The graph in figure 3 below is that of a one to one function since for any two different values of the input x (x _{1} and x_{2}) the outputs f(x_{1}) and f(x_{2}) are different.
The graph in figure 4 below is that of a NOT one to one function since for at least two different values of the input x (x_{1} and x_{2}) the outputs f(x_{1}) and f(x_{2}) are equal.
We can determine graphically if a given function is a one to one by drawing horizontal lines. If none of these horizontal lines cuts the graph of the function in two points or more the the function is a one to one; otherwise it is not a one to one.
## Examples of One to One Functions
## Example 1Show algebraically that all linear functions of the form f(x) = a x + b , with a ≠ 0, are one to one functions.SolutionWe use the contrapositive that states that function f is a one to one function if the following is true: if f(x _{1}) = f(x_{2}) then x_{1} = x_{2}We start with f(x _{1}) = f(x_{2})
which gives a x _{1} + b = a x_{2} + b
Simplify to obtain a ( x _{1} - x_{2}) = 0
Since a ≠ 0 the only condition for the above to be satisfied is to have x _{1} - x_{2} = 0
which gives x _{1} = x_{2}We have shown that f(x _{1}) = f(x_{2}) leads to x_{1} = x_{2} and according to the contrapositive above, all linear function of the form f(x) = a x + b , with a ≠ 0, are one to one functions.
## Example 2Show analytically and graphically that the function f(x) = - x^{2} + 3 is not a one to one functions.
SolutionWe use the contrapositive that states that function f is a one to one function if the following is true: if f(x _{1}) = f(x_{2}) then x_{1} = x_{2}We start with f(x _{1}) = f(x_{2})
which gives - x _{1}^{2} + 3 = - x_{2}^{2} + 3
Simplify to obtain - ( x _{1}^{2} - x_{2}^{2}) = 0
Factor - ( x _{1} - x_{2})( x_{1} + x_{2}) = 0
which gives
x _{1} - x_{2} = 0 or x_{1} + x_{2} = 0
which gives x _{1} = x_{2} or x_{1} = - x_{2}Since the above test does not strictly conclude that x _{1} = x_{2} the function is not a one to one.
Below is shown the graph of the given function and two horizontal lines are drawn: the x axis and the line y = - 2 (broken line) that shows clearly that there are two points of intersections and therefore the function is not a one to one.
## Example 3Show graphically that each of the following functions is a one to one function.1) f(x) = ln(x) 2) g(x) = e ^{x}3) h(x) = x ^{3}SolutionThe graph of each of the above functions is shown below with a horizontal line that shows one point of intersection only and therefore all the three functions are one to one functions. 1) 2) 3) ## How are One to One Functions Used?One to one functions are used in
1) Inverse
## More References and links on or related to one to one functionsQuestions on one to one Functionsinverse function definition properties |