|
Several questions with detailed solutions as well as exercises with answers on one to one functions are presented.
One to One Function
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.
Questions with Solutions
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 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.
- 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
One-To-One Functions
|