<< Chapter < Page Chapter >> Page >

Working rules

A. Onto function

  • A function is onto if every image in the co-domain has a pre-image in the domain set.
  • Range = Co-domain

B. Bijection

  • If a function is both one-one and onto, then the function is a bijection.

C. Inverse function

  • Solve given function for “x”.
  • Substitute “x” by inverse symbol “ f - 1 x ” and “y” by “x” to find the rule of inverse function.
  • Exchange the domain and range of the given function with that of inverse function.

Problem 1: A function f : R R is given by :

f x = sin 3 x + 2

Is the function surjective?

Solution :

Statement of the problem : Surjection is another name for onto function. Here, given function is a trigonometric function. We need to check whether its range is equal to co-domain.

According to question :

Domain = R

Co-domain = R

Now, let

y = f x = sin 3 x + 2

Solving for “x”, we have :

5 x + 2 = sin - 1 y

x = sin - 1 y 2 5

The inverse trigonometric function is valid in its domain of [-1,1]. Hence, values of “y”, for which “x” is real, is [-1,1]. Therefore,

Range = [ - 1 , 1 ]

Clearly, range is not equal to co-domain. Thus, given function is not an onto function.

Problem 2: A function f : R R is given by :

f x = a x 2 + 6 a 8 a + 6 x 8 x 2

Find the interval of values of “a” for which function is onto.

Solution :

Statement of the problem : It is given that the rational function is valid for all values of “x” as its domain is “R”. It is also given that co-domain of the function is “R”. In order to find the interval of "a", we consider that the function is an onto function. This implies that range of the function is equal to co-domain of the function i.e. “R”.

Let

y = a x 2 + 6 a 8 a + 6 x 8 x 2

a x 2 + 6 x 8 a y 6 x y + 8 x 2 y = 0

Rearranging as quadratic equation in variable “x”, we have :

a + 8 y x 2 + 6 1 y x 8 + a y = 0

Since “x” is real, discreminant of the quadratic equation should be non-negative,

36 1 y 2 4 X a + 8 y X 8 + a y 0

9 1 y 2 + a + 8 y X 8 + a y 0

Expanding and rearranging as quadratic equation in variable “y”, we have :

9 1 + y 2 2 y + 8 a + a 2 y + 64 y + 8 a y 2 0

9 + 8 a y 2 + a 2 + 46 y + 9 + 8 a 0

But, we have assumed that function is an onto function. As such, range of the function is equal to co-domain i.e “R”. It means that above inequality holds for all real values of “y”. Now, remember (as explained for sign scheme of quadratic equation) that a quadratic expression evaluates to non-negative number if (i) coefficient of “y2” is a positive number and (ii) descreminant is a non-positive number. Hence,

9 + 8 a > 0 a > - 9 / 8

and

a 2 + 46 2 { 2 9 + 8 a } 2 0

a 2 + 46 + 18 + 16 a a 2 + 46 18 16 a 0

a 2 + 16 a + 64 a 2 16 a + 28 0

a + 8 2 a 2 a 14 a + 28 0

a + 8 2 a 2 a 14 0

a 2 a 14 0

2 a 14

The first condition earlier yielded as a>= 9/8. The required interval, therefore, is intersection of two intervals, which is [2,14].

Problem 3: Let “f” be a one-one function with domain [x,y,z] and range [1,2,3]. It is given that only one of three conditions given below are true and remaining two are false :

f x = 1 ; f y 1 ; f z 2

Determine f - 1 1 .

Solution :

Statement of the problem : With three conditions, there are three different possibilities for one being true and other two being false. We shall check each such possibility to see whether function is onto function. If the function is onto, then the function is a bijection as it is already given that function is an injection.

Questions & Answers

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Functions' conversation and receive update notifications?

Ask