What are Functions?

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Functions are an important concept and are therefore, heavily tested on the new SAT. It is important to become proficient with them. But, what is a function anyway? In mathematics, a function relates to an output that resulted from an input. It is like in computer science when you learn about input and output devices and the relationship between both. For example, you put in an input into the computer in the form of instructions, and it resulted in an output in the form of a printed document. In the real world context, other ways in which the concept of functions occurs it in factories where you have large machinery. The machines work in the same context of input and output.

A function is usually denoted by “f(x) =…” , the f, is not unique and any letter can be used.

A function has three main parts:

  1. The input
  2. The relationship
  3. The output

Example: “Multiply by 4” is a very simple function you can grasp.

Input = 5, Relationship = × 4, Output = 20.

Consider this: For an input at 40 what is the output?

Examples of functions

f(x) = x2 + 3

g(x) = 2x – 7

The above definition of a function is considered as the basic understandable definition. It goes deeper. A function may also be considered as a relation that uniquely maps one member of a domain unto another member of a range which is a subset of a codomain. This means, if you have a function which maps set A to B, it is an object f such that   is mapped onto one or more members of the range such that. The range is part of a codomain. This means that a function can be a many to one or one to one relation. If you have a one to many relation that is not a function. Let us look at range, domain and codomain to get a better understanding of the aforementioned definition. But first let us look at one to one and one to many mapping.

Understanding the difference between the range and the codomain is very critical. It even becomes more critical when doing further math such as pre-calculus where a function is undefined.

Example 1

What is the domain and range of the function: f(x) = 2x2 + 6?

To answer this question, you need to think about all the possible values than can be inputted into the function and get an output. By looking at the function you realize than any real number can be placed into the function to produce an output. Also, you may also realize that the set of integers can also be used. If you use any random integer for example -3, you can check

Thus f(-3) = 2(-3)2 + 6

= 2(9) + 6

= 18 + 6

= 22

You may also notice that all the output for the set of real numbers will be the same for the set of integers because of the x2. Therefore the domain is the set of integers. Since the set of real numbers is a subset of the set of integers, we can exclude it. Note that the domain does not have one set definition, because it could also be the set of counting number, whole numbers, etc. However for the range, you will never get a negative output, so the range could never be the set of integers, but it could possibly be the set of even numbers, however the range is the set {8, 14, 22…}. Though the question didn’t ask, but the codomain could be the set of counting numbers, since the output could be any possible counting number.

Evaluating Functions

Evaluating functions is quite simple, all you need to do, if plug in the value that is given, substitute the value for I, then solve the equation.

Example 2

Function f is defined by
f(x) = – 2x2 + 6x – 3, find f(- 2).

What you do in this instance, you substitute -2 for ‘x’ in the function then solve

= -2(-2)2 + 6(-2) – 3

= -2(4) – 12 – 3

= -4 – 12 – 3

= – 19

Example 3

Two functions f and g is defined by

f(x) = 2x + 7 and g(x) = -5x – 3, find (f + g)(x).

When you look at the question it might seem hard, but it is quite simple. You may note that (f + g)(x) denotes is the product of two numbers ideally. It can be expanded as such

(f + g)(x) = f(x) + g(x). As such, this is in a form that we can understand therefore it is:

f(x) + g(x) = (2x + 7) + (-5x – 3). Note: the brackets are there to show the two distinct functions

2x + 7 – 5x – 3 by grouping like terms”

2x – 5x + 7 – 3

= – 3x + 4

Therefore (f + g)(x) = – 3x + 4

Example 4

Two functions f and g is defined by

f(x) = 2x2 – 3x and g(x) = x + 1, find f (g)(-3).

What you see above is called a function of a function. It is quite easier than it looks. What is means that wherever you see x in function f, you are going to plug put the g function then plug in -3. Therefore:

f (g)(x) = 2(x + 1)2 – 3(x + 1). Thus, we first put the g function in the f function. Now we evaluate for:

f (g)(-3) = 2(-3 + 1)2 – 3(-3 + 1)

= 2(-2)2 – 3(-2)

= 2(4) + 6

= 8 + 6

= 14

Therefore f (g)(-3) = 14

If we were not asked to evaluate f (g)(-3), but just find f (g)(x)

The answer would just be: 2(x + 1)2 – 3(x + 1).

You may also note that a function can be graphed and in that context, the x is the domain and y is the range so sometimes you will see that f(x) = y.

Range, Domain and Codomain

The first diagram represents a one to one mapping, where each element of X maps exactly onto one Element of Y. The second diagram represents aone to many mapping where one element of X maps onto one of more element of Y.

A domain is simply all the possible values that can be inputted into a function to produce an output.

The range is simply all the output of a function.

The codomain is all the possible outcomes or output from a function.

 

For more awesome SAT math concepts review, visit our blog: http://exammasters.ca/category/math/

For a FREE SAT Math Practice Test: http://exammasters.ca/free-sat-math-practice-test/

 

How to Make Sure Your Child Sticks to Good Habits This Summer

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For kids, summer is a time to slack off, goof off, and generally forget they have any sort of responsibilities whatsoever. And yes, while there is room for some of this, it’s the job of parents to make sure their children stick to good habits during the free-for-all that is summer break. Without some sort of schedule, kids can waste valuable time, fall behind in their education, and develop risky behaviors. Here’s how you can make sure your child maintains safe, productive habits.

 

Schedule time to learn/improve at a hobby

 

Instead of just setting a strict summer schedule for the sake of fitting in time to read, do chores, and setting consistent sleep patterns (all of which are important), use a summer schedule to build in time for self-improvement. Learn or get better at a task or hobby.

 

“If you love doing something enough, you should find time to do it. It will make you happier and healthier, which means you’re bound to do better in school and everything else as well,” notes HerCampus.com “Set aside a time every few days or every week, for instance, to pursue your newly found hobby. By working it into your schedule, it won’t feel like something you can do without,”

 

Setting goals for the summer is a great way to make sure kids stay occupied in a productive manner. One or two summer’s end goals is probably enough – you don’t want to overdo it and spread your child too thin. Summer still needs to feel like summer.

 

Prevent education stagnation

 

Though you want a good part of the summer to be about outdoor activities, sports, games, pool time, and just lounging around, you would be doing your child a disservice not to schedule in some sort of educational activity almost every day. Whether your kid needs help on a specific school subject they’re struggling with, or you simply want to give them a kickstart for the next school year, there are plenty of ways to fit education into the summer schedule – and a lot of it can even be fun.

 

  • Schedule reading hours every day. This one is a no-brainer. Have you child sit down and read a book every day of the summer. Coordinate with their school to see if they can get a head start on any of the upcoming year’s reading materials.
  • Take educational trips. In most cities/towns in the US you’re never too far away from some sort of museum, national landmark, or historical site of significance. Schedule day trips and even longer vacations with education in mind.
  • Get your child involved in summer meals. “Cooking is a great way to teach about measurements and how to use fractions. Theme nights for dinner not only teach cooking skills, but math, as you teach your child how to measure and convert weights, volumes and numbers,” notes The Huffington Post.

 

Talk to your child about the dangers of substances

 

Summer, with all its free time, increases the chances of your child experimenting with or falling into patterns of risky behavior. You must keep an open communication line when it comes to drugs and alcohol. Your kid needs to know that they can be honest with you without fear of unjust punishment, yelling, or any other general overreaction. Use books, TV, and movies as segways into frank discussions.

 

“Take advantage of ‘teachable moments’. If you see a character in a movie or on TV with a cigarette, talk about smoking, nicotine addiction, and what smoking does to a person’s body. This can lead into a discussion about other drugs and how they could cause harm,” says one good suggestion.
If you want to make sure your child sticks to good habits this summer, you need to set a schedule, occupy their time, and focus on fun sources of education and other out-of-the-box learning experiences. On top of that, you need to be frank about the dangers of substances, which can increase as your child spends more free time with friends during the summer months.

 

Author: Laura Pearson

Another Proud Moment! 

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by Staff Contributor 

Just in time for National Poetry Month! All 4 One Tutoring is very excited and proud of our CEO, Dr. Charice Hayes. With the hectic role of running All 4 One Tutoring, she also takes time not only to teach others but write books. How exciting is that!!!

 Dr. Hayes just published a short book of empowering poems and quotes titled To Get You Through. This books encapsulates inspiring quotes and poems to keep you motivated and on path to whatever are your goals and desires. You can purchase the book at Barnes and Noble online. You can even download it in your Apple iBooks.

Dr. Hayes is also the author of two children’s books Tommy and His Cars and Tomasito y Sus Coches. These books can also be purchased in iBooks and barnesandnoble.com

We’re Semi-finalist! 

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by staff contributor 

All 4 One Tutoring is so grateful to be semi-finalist in Idea Cafe’s small business grant. We have made it through a very competitive initial round of selection by a panel of expert judges to advance to the semi-finalist. This is a very exciting time for us!
Please make sure to support and vote for us. The grant money will allow us to expand and reach others in order to build strong lives. Go to http://www.businessownersideacafe.com/18th-small-busines-grant/voting?profile=17809 to cast your vote. 

Blended Learning

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by Dr. Charice Hayes 


What is blended learning? 

Blended learning is traditional teaching combined with digital media or online learning. I like to also refer to this type of learning as “hybrid” learning. 

Why use blended learning?

  • Research suggests that blended learning is more effective than learning in a traditional setting. 
  • Blended learning improves digital literacy, along with online learning skills. 
  • Some teachers/professors are using flip learning
  • Blended learning aligns with college and workplace readiness. 

How can you transform a traditional classroom into a blended classroom? 

  • You can use online videos to supplement instruction. 
  • You can use the flipped learning model.
  • You can incorporate chats and discussion boards. 
  • You can use software such as Socrative.com.

What are your thoughts on blended learning? Please share your thoughts. 

2016 Year in Review 

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Contributed by Dr. Charice Hayes CEO


It has been a continued promising year for All 4 One Tutoring. This past year, we’ve put some things in place to diversify and expand our company and programming. Within the first five months of 2016, we were able to hire tutors and instructors in our after-school program. In November, we’ve started the process of hiring a marketing intern. This additional hiring helped us grow and continue to mirror the company’s mission statement. We’re looking forward to hiring more staff members in the first 3-4 months of 2017. 
In the summer of 2016, we continued a summer tutoring program. Some of our clients wanted to avoid that dreadful “summer learning loss.” We conducted online and face-to-face sessions. We plan to do the same summer 2017. 

During the summer of 2016, All 4 One Tutoring had to close one of its offices due to space. We are currently looking for a larger office (That’s a good thing!) We still have office space at The Empowerment Academy

Our after-school program continues to get praise by administrators, community leaders, and parents. This school year, we continued with more project-driven instruction and projects. We have even purchased state of the art headphones for our learning labs.

We are looking forward to finishing the 2016-2017 school year with continued success. 

Last but not least, for a fourth year in a row, All 4 One Tutoring‘s income and net profit has increased. Our income has increased 40% from 2015, and our net profit has increased 30%. We would love to see this trend occurring year after year. Being the owner of such a blossoming and empowering organization is such an honor. We are proof that when we work together to achieve a shared goal, we can expand quality education and training so that everyone can strive for positive social change. We have more plans underway for 2017!!!!
  

Here’s a look at All 4 One Tutoring’s agenda in 2017:
-more partnerships (international and nationwide)

-opening a bigger office

-increasing staff 

-offering more training and workshops 

-revisiting 21st CCLC (Century Community Learning Centers)

-expanding our foreign language service

-developing an internship program 

I’d like to thank everyone who has had a positive impact and who has supported All 4 One Tutoring this year previously, and I always look forward to your continued support. Together we can reach the goal of educational EMPOWERMENT and provide positive SOCIAL CHANGE.

Dr. Charice Hayes, CEO