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Questions on one to one Functions

Several questions with detailed solutions as well as exercises with answers on one to one functions are presented.

From the definition of one-to-one functions we can write that a given function f(x) is one-to-one

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.

    graph of function f in question 3


  • 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 one-to-one 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.

  1. f = {(12 , 2),(15 , 4),(19 , -4),(25 , 6),(78 , 0)}

  2. g = {(-1 , 2),(0 , 4),(9 , -4),(18 , 6),(23 , -4)}

  3. h(x) = x 2 + 2

  4. i(x) = 1 / (2x - 4)

  5. j(x) = -5x + 1/2

  6. k(x) = 1 / |x - 4|


Answers to Above Exercises

  1. f is a one to one function

  2. g is not a one to one function

  3. h is not a one to one function

  4. i is a one to one function

  5. j is a one to one function

  6. k is not a one to one function
Interactive tutorial on One-To-One Functions


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Updated: 2 April 2013

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