Several questions with detailed solutions as well as exercises with answers on one to one functions are presented.
From the definition of onetoone functions we can write that a given function f(x) is onetoone
if A is not equal to B then f(A) is not equal f(B)
where A and B are any values of the variable x in the domain of function f.
The contrapositive of the above definition is as follows:
if f(A) = f(B) then A = B
where A and B are any values of x included in the domain of f. We will use this contrapositive of the definition of one to one functions to find out whether a given function is a one to one.
Question 1: Is function f defined by
f = {(1 , 2),(3 , 4),(5 , 6),(8 , 6),(10 , 1)},
a one to one function?
Solution to Question 1:
 Two different values in the domain, namely 5 and 6, have the same output, hence function f is not a one to one function.
Question 2: Is function g defined by
g = {(1 , 2),(0 , 4),(2 , 4),(5 , 6),(10 , 0)},
a one to one function?
Solution to Question 2:
 Consider any two different values in the domain of function g and check that their corresponding output are different. Hence function g is a one to one function.
Question 3: Is function f given by
f(x) = x^{ 3} + 3 x^{ 2}  2,
a one to one function?
Solution to Question 3:
 A graph and the horizontal line test can help to answer the above question.
 Since a horizontal line cuts the graph of f at 3 different points, that means that they are at least 3 different inputs x1, x2 and x3 with the same output Y and therefore f is not a one to one function.
Question 4: Show that all linear functions of the form
f(x) = a x + b,
where a and b are real numbers such that a not equal to zero, are one to one functions.
Solution to Question 4:
 We start with f(A) = f(B) and show that this leads to a = b
a(A) + b = a(B) + b
 Add b to both sides of the equation to obtain
a(A) = a(B)
 Divide both sides by a since it is not equal to zero to obtain
A = B
 Since we have proved that f(A) = f(B) leads to A = B then all linear functions of the form f(x) = a x + b are onetoone functions.
A = B
Question 5: Show that all functions of the form
f(x) = a (x  h)^{ 2} + k , for x >= h ,
where a, h and k are real numbers such that a not equal to zero, are one to one functions.
Solution to Question 5:
 We start with f(A) = f(B)
a (A  h)^{ 2} + k = a (B  h)^{ 2} + k
 Add k to both sides of the equation to obtain
a (A  h)^{ 2} = a (B  h)^{ 2}
 Divide both sides by a since it not equal to 0
(A  h)^{ 2} = (B  h)^{ 2}
 The above equation leads to two other equations
(A  h) = (B  h) or (A  h) =  (B  h)
 The first equation leads
A = B
 Let us examine the second equation. The domain of f is all values of x such that x >= h. This leads to x  h >= 0 which in turn leads to A  h >= 0 and B  h >= 0 which means the second equation (A  h) =  (B  h) does not have a solution.
Question 6: Is function f given by
f(x) = 1 / (x  2)^{ 2},
a one to one function?
Solution to Question 6:
 It is easy to find two values of x that correspond to the same value of the function.
f(0) = 1 / 4 and f(4) = 1 / 4. For two different values in the domain of f correspond one same value of the range and therefore function f is not a one to one.
Question 7: Show that all the rational functions of the form
f(x) = 1 / (a x + b)
where a, and b are real numbers such that a not equal to zero, are one to one functions.
Solution to Question 7:
 Let us write an equation starting with f(A) = f(B)
1 / (a A + b) = 1 / (a B + b)
 Multiply both sides of the equation by (a A + b)(a B + b) and simplify
a B + b = a A + b
 Add b to both sides
a B = a A
 Divide both sides by a to obtain
B = A
 The given functions are one to one functions
Exercises For each of these functions, state whether it is a one to one function.
 f = {(12 , 2),(15 , 4),(19 , 4),(25 , 6),(78 , 0)}
 g = {(1 , 2),(0 , 4),(9 , 4),(18 , 6),(23 , 4)}
 h(x) = x^{ 2} + 2
 i(x) = 1 / (2x  4)
 j(x) = 5x + 1/2
 k(x) = 1 / x  4
Answers to Above Exercises
 f is a one to one function
 g is not a one to one function
 h is not a one to one function
 i is a one to one function
 j is a one to one function
 k is not a one to one function
Interactive tutorial on
OneToOne Functions
