What is a Function? | TASC Math

Businessman drawing bar chart and other infographics in note pad

Functions relate an input to an output.

In mathematical equations, a function is typically written as “f(x) = …” There are other ways to express functions, but you should keep this expression in mind; functions are a high emphasis area on the TASC Test Assessing Secondary Completion™ Math subtest.

There are many things you can do with a function and many different ways to express functions. Here, we’ll focus on the basics.

First, you’ll need to know about relations. According to Elizabeth Stapel of Purple Math, a relation is “a relationship between sets of information. Think of all the people in one of your classes, and think of their heights. The pairing of names and heights is a relation. In relations and functions, the pairs of names and heights are “ordered,” which means one comes first and the other comes second.”

The set of all the starting points (the numbers that come first) is called the domain, and the set of all the ending points (the second set of numbers) is the range. The domain is what you start with, and the range is what you end with – or, to put it another way, the domain is the x and the range is the y.

Stapel calls functions “well-behaved relations” because when you are given a starting point, you know exactly where to go. Functions are single-valued. Whenever you’re given the x, you can find the y.

You may see functions expressed in a chart, such as:

X: x Y: x2
0.5 0.25
2 4
5 25


You may also see them expressed in ordered pairs. In these pairs, the input is listed first and the output is listed second: (input, output). Or, put differently: (x, f(x)).

A function can then be defined as a set of ordered pairs. For example, a function can be expressed in this way: {(3,5), (6,8), (11,4)}

This example says 3 is related to 5, 6 is related to 8, and 11 is related to 4. The domain is {3,6,11}. The range is {5,8,4}.

Functions are similar to graphs and inequalities, so keep those tips and tricks in mind while you’re working through these functions, and keep practicing.

Forces and Interactions | TASC Science


Forces and their interactions can be complicated, and difficult to navigate – especially if physics isn’t your best subject. When you’re preparing for the TASC Test Assessing Secondary Completion™ Science subtest, familiarize yourself with important physical equations relating to force however. These equations can help you solve for important factors, such as the forces between objects, gravitational pull, and electrostatic force.

At McGraw-Hill Education CTB, we know you’re working hard to familiarize yourself with every aspect of the TASC test. To help you work through the Science subtest, we’ve outlined two significant laws related to forces and their interactions: Newton’s Law of Gravitation and Coulomb’s Law. Add this new information to your science study tips – and keep working towards your high school equivalency.

Newton’s Law of Gravitation

You might be familiar with Sir Isaac Newton – many people tell the story of an apple falling on Newton’s head, helping him discover gravity. Not only is Newton one of the most famous scientists in our history, but he also continues to be influential today.

His theories, like the Law of Gravitation, are fundamental to physical law. Newton first published this law in 1687. He hypothesized that every body in the universe is attracted to every other body with a force that is directly proportional to the product of the bodies’ masses and inversely proportional to the square of the bodies’ separation. This may sound complex, but we experience this effect everywhere on the planet; gravity that is responsible for holding us to the earth.

The Law of Gravitation can be expressed in a mathematical formula:

Fg = G(m1m2/r2)

In other words, Newton’s Law of Gravitation states that the force of gravity (Fg) between two particles of mass (m1 and m2) has a specific magnitude, which can be discovered when the above equation is solved (with r as the distance between the center of the two masses and G as the gravitational constant).

The Next Generation Science Standards expect students to be able to “describe and predict the gravitational and electrostatic forces between objects” using Newton’s Law, and Coulomb’s Law. To do this, you might also want to know that Henry Cavendish determined the value of G experimentally in 1798. The value of G can be expressed as:

G = 6.67 x 10-11N x m2/kg2

Coulomb’s Law

Coulomb’s Law focuses on the interaction between charged objects. This is a non-contact force that acts over some distance of separation. Regardless of what objects are being attracted to each other, there are always two charges and a distance between them – and these are the three variables that influence the strength of the interaction.

Relying on the three variables on electric force, Coulomb’s Law states that the electrical force between two charged objects is directly proportional to the product of the quantity of charge on the objects and inversely proportional to the square of the separation distance between the two objects.

Coulomb’s Law can be expressed in a mathematical formula:

F = (k • Q1 • Q2)/d2

Here, Q1 represents the quantity of charge on object one, and Q2 is the charge on object two. The variable d represents the distance of separation between the two objects, in meters.

The variable k is a proportionality constant known as Coulomb’s law constant. It depends on the medium in which the charged objects are immersed. For example, the value for air is approximately 9.0 x 109N x m2/C2. The value for water, on the other hand, can be reduced by as much as a factor of 80. It is worth noting that the units of k, when substituted into the equation, will cancel the units on charge and the units on distance – and leave only a Newton, or unit of force.

Lastly, the direction of the electrical force depends on whether the charged objects are charged with a like charge, or with an opposite charge. It also depends on the spatial orientation. There are fundamental rules of charge interactionthat can help you determine the direction of the electrical force. However, by knowing the type of charge on the two objects, you can accurately determine the direction of the force. For example, when two objects both have a like charge, they direct away from one another.


You can use the mathematical equations of Newton’s Law of Gravitation and Coulomb’s Law to describe – and predict – forces between objects. Forces and interactions are a medium emphasis topic on the TASC Science subtest. Some tests could have questions that deal with force, and some might not. Either way, you want to be prepared. Keep these laws in mind while you are practicing for the TASC test.

Using the Properties of Exponents | TASC Math

hand written maths calculations

An exponent is a small number that appears in the upper right-hand corner of a base. Sometimes, an exponent is referred to as a power or index. This number says how many times to use the base in a multiplication. For example:

23 = (2 x 2 x 2) = 8

Exponents can be tricky to deal with, but only if you’re unfamiliar with them. Knowing the properties of exponents, and how to use them to rewrite expressions – like radical expressions – can help you better navigate the TASC Test Assessing Secondary Completion™ Math subtest.

The Rules of Exponents

There are three keys to understanding exponents:

  1. The exponent says how many times to use the number in a multiplication. This, you’ll know, is the basic definition of exponents.
  2. A negative exponent means divide, because division is the opposite function of multiplying.
  3. A fractional exponent, like 1/x, means to take the xth root. For example: x1/2 = 2√x

If you understand these three keys, you can essentially work through any exponent because all the other rules of exponents are based on these. Keep this in mind as you look at the following properties of exponents.

Product of Powers Property

You may be asked to solve an equation such as: 52 x 58. You could solve for each exponent, and then multiply the two numbers. Or, you can take a shortcut. Because of the properties of exponents, you can simply add the exponents together. For the above example, the answer is: 510.

When an equation asks you to multiply two powers with the same exponent but different bases, you should multiply the bases (like normal) and keep the exponent. For example:

ax x bx = abx

23 x 63 = 123


Usually, you can leave answers in this exponent form. However, if you don’t notice this answer in a multiple-choice question then you should solve for the exponent. Remember, all you need to solve for the exponent is to multiply the number by itself as many times as indicated by the exponent.


Zero Exponents


Any base that is raised to the power of zero is 1. That means all of these exponents equal 1:


30 = 1

2,8450 = 1

x0 = 1


This may seem strange to you – especially because you know any number multiplied by zero equals zero. The complex answer is that for all real numbers x, x ≠ 0.


It should be noted that 00 is undefined.


Power to Power


You may be asked to take the power of a power. In this case, you will multiply the exponent in the base by the exponent that is acting on the base. For example:


(34)2 = 34×2 = 38


The steps are slightly different if you raise a product to a power. In this case, you are actually raising each factor to a power:


(2x)2 = 22x2


 Quotient of Powers Property


Just as you can multiple exponents, you can also divide exponents. When you divide powers with the same base, you simply subtract the exponents like this:


26 = (2 x 2 x 2 x 2 x 2 x 2) = 22

24           (2 x 2 x 2 x 2)         .


Here, we’ve crossed out a 2 for each 2 on the bottom to demonstrate what the subtraction of exponents actually looks like. Even more simply, you can subtract 4 from 6.


If you see a quotient raised to a power, you must distribute the exponent to both the numerator and the denominator:


(x/y)3 = x3/y3


Simplify Radical Expressions


Using what we know about the properties of exponents – such as the third key above and the quotient of powers property – we get the product property of radicals, and the quotient property of radicals. These two properties demonstrate that the square root of a product equals the product of the square roots of the factors. For example:


√2x = (√2 x √x)


In radical expressions, the answer cannot be negative; we wouldn’t be able to solve for a real answer if it was.


When you simplify a radical expression, you know it is in its simplest form when there are:

  • No perfect square factors other than 1 in the radicand (or division symbol)
  • No fractions in the radicand
  • No radicals in the denominator of a fraction


If the denominator is not a perfect square, then you can rationalize the denominator by multiplying the expression by an appropriate form of one. Check out examples of these radical expressions, and practice simplifying a radical expression.

Round and Flat Characters | TASC Reading

stack of books

In his book Aspects of the Novel, British novelist E. M. Forster first used the terms round and flat to describe characters. According to Forster, round characters have “the incalculability of life” about them. Round characters are complex, like actual human beings, and can surprise readers. The teachers at Education Portal  note of round characters, “When you think you’ve figured them out, they can do something that catches the reader off guard.”

Main characters are typically round characters. As Brandi Reissenweber, a writing expert at the Gotham Writers Workshop, states in her recommendations to writers, round characters demonstrate dimensionality because there are multiple facets of the character’s personality. In other words, a round character is multidimensional. Round characters are sometimes called dynamic, meaning they undergo change (most often progress) and are complex. In a story, play, or novel, a round character demonstrates this change in a variety of ways. Dynamic characters experience this change as a result of action in the plot, and this change is significant enough to make a permanent change within the character beyond his or her mood.

In contrast, flat characters are not complex or dynamic. They do not change, and may be referred to as static. Often, they are not even interesting to readers. You can recognize a flat character because he or she can be described in a sentence or two, and is often aligned with a type or cliché. As the teachers note, “They’re no problem to figure out, which is probably why many blockbuster action flicks are stacked with flat characters.” Flat characters never surprise the reader.

Brushing up on these types of characters can help you prepare for the TASC Test Assessing Secondary Completion™ Reading subtest. Round and flat characters are a medium emphasis  topic, and you may be asked questions about characters from reading passages. Knowing what types of characters they are, and the significance of their actions, decisions, or transformations, can help inform your answers. The trick to remembering these types of characters is easy: the more human characters feel, the rounder they are.

Answering Your Top 10 FAQs About the TASC Test

Meaning of life


  1. How do I know if I passed?

Check with your testing center or State Department of Education to find out how to check the results of your TASC Test Assessing Secondary Completion. Most states use an online portal to display the status of your high school equivalency exam. If the status of your test reads “Diploma” or “Passing Transcript,” you passed.


Your state is responsible for awarding you with your high school equivalency certificate if you’ve passed the TASC test and met any additional requirements.


  1. When do I get my results? Can I see my scores online?

Each state has different policies and timelines for sharing score results with examinees. After McGraw-Hill Education CTB receives paper-and-pencil-based tests in the mail, it can take up to 10 business days for reports to load into the reporting system. If bubbling mistakes are made on the answer sheet, however, you can expect up to a 6-week delay before seeing your results. For computer-based tests, reports are added to the system within 24 hours.


You can obtain your results from your testing center. Typically, test centers forward results and issue diplomas within a day or two of obtaining the results from CTB. There is not yet a CTB-based online portal to check your scores or diploma status, so contact your test center for your full score report.


If you’re concerned with the whereabouts of your test results, contact your test center. You can request they provide you with a Candidate Report, which shows your preliminary results and scores on the subtests (except the Writing subtest). If your scores are not posted after 10 business days, you can call TASC test Customer Support at 888.282.0589 from 9 AM to 7 PM Eastern Time.


  1. How does scoring work?

To pass the TASC test, you must score at least 500 out of 800 on each subject test, along with 2 out of 8 on the Writing prompt. If you don’t pass specific subject tests, you can retake those subject tests individually.


Passing scores for each TASC subject test were set using a nationally representative sample of high school seniors with norm-referenced information. The scores were validated through correspondence of cut scores to the 2002 GED® test series cut scores and the percentage of adult examinees expected to pass the TASC test, based on data from the field test.


  1. How does the TASC test compare to the previous GED test?

CTB designed the TASC test after seeing a need for an affordable test that aligns to current educational standards. The previous GED test covered outdated content that didn’t align with the College and Career Readiness Standards. The TASC test gradually transitions examinees to the new national standards, focusing on preparing adults for careers or higher education rather than requiring them to memorize facts to pass a test.


  1. What is on the Math subtest of the TASC test?

You should concentrate heavily on building your mathematical reasoning skills as you prepare to take the TASC Math subtest. Math has a reputation of overwhelming students, but any student can acquire the skills needed to pass the TASC test. You can learn to recognize and understand the building levels of mathematics – and that’s mathematical reasoning. Check out these resources as you prepare:

  1. What resources are available in Spanish?

There is a Spanish version of the TASC test, along with Spanish practice questions:

  1. Will you publish any additional preparation materials in the future (Spanish Readiness exam, additional readiness exam, Spanish version of the preparation book, etc.)?

Most study materials are published by organizations separate from CTB. Feel free to contact them with your questions as you search for the high school equivalency resources that are best for you.

  1. How do I find a TASC test center in my state?

If your state offers the TASC test, there are several testing centers you can access. Select your state to search for the center nearest you.

  1. What is the test schedule when the TASC test is taken over two days?

If your TASC test is scheduled over two days, your test center determines the order of the subject tests. Contact your center to find out the order so you can prepare for test day.

  1. Can homeschooled students take the TASC test to earn a diploma?

The TASC test was designed not only for adults who have not graduated from high school but also anyone 16 years of age or older who are not enrolled in a traditional school. Please note that the TASC test qualifies you for not a high school diploma but a high school equivalency, which is accepted as the equivalent to a high school diploma by most colleges and employers.

Do you have any other questions about the TASC test? Explore our FAQs page, and contact us if there’s anything else you want to know.

4 Tips for Improving Your Spelling | TASC Writing


It might seem like an insignificant skill, but spelling can dramatically improve your writing. Though spelling and grammar are not as important as constructing a clear and persuasive argument, these skills are still important – and should be practiced. Especially for the TASC Test Assessing Secondary Completion™ Writing subtest, youwant your grammar and spelling to be as close to perfect as possible.

There are steps you can take during the studying process to help you improve. Additionally, there are some quick tips you can add to your test-taking tool bag to improve your spelling.

1. I before E, except after C and when sounding as “a” as in neighbor and weigh.

Most people have heard this spelling rule before. However, the rule is trickier than it seems. It always works for words such as achieve, believe, yield and conceit, deceive, perceive, feign.

What most people don’t know is that the rule only applies when the sound represented by “ie” or “ei” makes the long e sound (/ee/). For example, the rule does not work for the word science where the I is before the E despite the fact that they follow C because the “ie” does not make the long e sound.

Additionally, as the experts at the Oxford Dictionary point out, the rule also does not apply to any word without the long e sound, even when there is no C. In this case then, words that have the long e sound always have the “ei” form rather than the “ie” form. Examples include foreign, height, vein, and weight.

The Oxford Dictionary points out that there are very few exceptions to this, such as seize, weird, and caffeine. These exceptions must be memorized.

2. Forming Adverbs

While you are writing, you may want to use adjectives as adverbs to describe your verbs. Adverbs are easy to form, when you know the trick.

The most basic rule is that you must add –ly to the end of the adjective. For example, sudden becomes the adverb suddenly when you add the –ly.

If the adjective has two syllables and it ends in –y, then you must replace the –y with an –i before you add –ly. For example, happy becomes happily.

If the adjective ends with a consonant followed by –le, replace the final –e with –y on its own (because the –l is already there). Incredible becomes the adverb incredibly, for example.

If an adjective ends with –ly, like lively, then it cannot be made into an adverb. You must use a different form of the word, or a synonymous word.

3. Adding Endings to Words that End in -Y

One of the best ways to expand your vocabulary is simply to know how to change the words you already know into different forms. Like the adverb rule above, it’s easy to change words that end in –y into adjectives and other verb forms.

When a word ends with a consonant and –y, you can change the final –y to an –i and add an adjective ending. For example, pretty becomes prettier or prettiest. Similarly, beauty becomes beautiful.

The same rule applies whenever you want to change the verb form to present, past, or future when the verb ends with -y. For example, apply becomes applies, applying, or applied.

4. More on Verb Tenses

The basic form of a verb is called the infinitive. It is not conjugated and is often written with the word to before it. If you know the infinitive, you can change that verb to any tense you want. To make the past tense of regular verbs, you add the ending –ed to the infinitive. To put a verb in present participle, you add the ending –ing to the infinitive. Often, you do not need to make any other spelling changes. For example:

to borrow



However, there are some tricks for adding these endings to words that end in vowels. For example, when a verb ends with a silent e, then you must drop the silent e and then add –ed or –ing.

to rake



When a verb ends in –ee, -ye, and –oe, you must leave the final –e whenever you are adding –ing. For example:

to free



If the verb ends in a single vowel and a consonant (that is stressed, or if the word is a single syllable), then you must double the final consonant before you add –ed or –ing. You do not double the consonant if it is not stressed, nor do you double it if there are two vowels before the consonant. For example:

to refer



When a verb ends in –c, you need to add a –k before adding –ed, -ing, or –er.

to mimic




Want to test yourself? See if you can put these tips to the test with spelling quizzes from the Capital Community College Foundation.

Real Talk: Travis Dambrot’s Success Story

travis diploma

Twenty-four-year-old Travis Dambrot recently passed the TASC Test Assessing Secondary Completion™ in New York. Find out how, after dropping out of high school in 10th grade, he overcame anxiety issues to earn his high school equivalency.
Prior to taking the TASC test, what was the highest-grade level you completed (and how long ago was it?)
I don’t remember 100%, but I believe it was 10th grade and I think it was back in 2008. For the people who do the math, I realize that I should have graduated that year but I was held back twice, once in second grade due to my inability to read at the time, and again in ninth grade because of issues with anxiety that I had. I suffered from agoraphobia and panic disorder, which is the reason I dropped out in the first place.

What’s your inspiration for pursuing your high school equivalency (HSE)?
Once I overcame my anxiety disorders earlier this year, getting my HSE was a top priority. It took me several months just to get an ID, but once I got that, everything else fell into place.

Did you take the TASCtest online or at a testing center?
I took the paper-and-pencil version at my local testing center.
How did you prepare for the TASC test? How did the study materials help you prepare for the TASC test?
I mostly used online resources. I can’t recommend The KHAN ACADEMY® enough. Definitely check them out. I also used Wikipedia and watched a ton of history-related documentaries on YouTube™.
Which sections or subjects did you find to be the easiest and also most challenging?
Social Studies was the easiest for me. I’ve always been fond of history, geography, and civics. The hardest was math, which is funny because it was probably my favorite subject back in grade school, but it has definitely gotten way more challenging since then!

How do you feel now that you took the TASC test and received your high school equivalency?
It’s incredible. I was expecting to ace Social Studies and maybe pass Reading and Writing, but I never imagined passing Math or Science, so it’s really exhilarating!

What is your number one piece of advice or tip for all TASC test takers?
That’s a good question. I guess I’d recommend that you shouldn’t underestimate yourself. Like I said before, I didn’t think I’d pass the test, but I did. It’s my understanding that they are getting rid of the multiple-choice within the next couple of years, but as it stands right now, there is no harm in guessing if you don’t know an answer.


Have you passed the TASC test? Share your story with us.

Matter and Its Interactions | TASC Science

Periodic Table of Elements

There are many aspects to understanding matter and its interactions, according to the Next Generation Science Standards. Students preparing for the TASC Test Assessing Secondary Completion™, for example, will want to be familiar with the periodic table of elements (pictured below).

Students might be asked to compare different elements. To be able to do that, students will need to know the characteristics of elements – in other words, the characteristics of matter. This knowledge can help test takers make comparisons and observe matter’s interactions through the use of simple chemistry.

There are three states of matter.

These three states are: solid, liquid, and gaseous. Solids are rigid. They have a definite volume and shape, which is often unchanging or difficult to change. Liquids and gases are more fluid, and do not have a particular form or shape. Rather, both take the shape of the container they are placed into (to some extent).

It is possible for some substances to change states. According to Tanner’s General Chemistry,  “substances can be transformed from one state to another by heating or cooling.” Additionally, changes in pressure can change the substance’s state as well.

For example, water can exist in any of the three states. In it’s most common state, water is a liquid. When water is frozen, it becomes a solid – ice. Water is a gas when it evaporates.

Think of these three states in terms of molecules. In solids, molecules are very close together. In liquids, molecules are more spread out but not as spread out as in gases.

Molecules are the smallest bits of compounds. It’s important that students distinguish molecules from particles, which are the building blocks of matter.

There are two kinds of matter.

The first kind is material, which itself can be divided into two categories. The first is a material that has a consistent composition throughout. Because of this consistent composition, these types of matter are often described as homogeneous. Examples include metal, plastic, and even paper. The second, then, is heterogeneous, which means it’s varied in composition. Examples of a heterogeneous material include particleboard and wood grains.

The second kind is a substance, which has a rather definite chemical composition and is therefore always homogeneous. Substances usually refer to elements that have definite rations of components, including salt and nitrogen for example.

Matter interacts in four ways.

According to the Lawrence Berkeley National Laboratory’s Particle Data Group,  there are “four fundamental interactions between particles, and all forces in the world can be attributed to these four interactions.

These four fundamental interactions, or forces, are:

  1. Strong: A force that holds the nucleus together. It counters the forces of repulsion of protons.
  2. Electromagnetic: A force that is created between charges and magnetic force. It depends on an exchange of photons.
  3. Weak: A force that is only effective at very short distances. It’s usually discussed in terms of beta decay.
  4. Gravity: A force that is always attractive and acts along the line joining the centers of mass of two pieces of matter. It’s similar to an electromagnetic force, but it’s the weakest of the four forces.

4 Study Tips for Making Thanksgiving Productive | TASC Test

The holidays can be a stressful – and wonderful – time of the year. Whether you’re having family from out of town in your home, or you’re traveling yourself, it can take a lot of planning. (And that doesn’t even include cooking time.)

If you’re preparing for the TASC Test Assessing Secondary Completion™, you might feel like you don’t have time for it all. To help you balance everything, take these study tips for making your holiday more productive:

1. Over-Prepare

As the instructors at Western Governors University suggest, one of the best tips for making your holiday productive is to make a detailed schedule. This schedule should not only make time for studying, but should also make time for your Thanksgiving activities.

We suggest using either a day planner, or a digital calendar like iCal or Google Calendar. Both forms provide you with a format to break a single day into blocks of time. In this way, you can map out a two-hour block of time to study before you have to help with Thanksgiving dinner or an hour review session before the football game.

This might mean that you have to get up a little earlier than usual to study in the morning; or it could mean that you can’t do all of the holiday activities this year. But as the instructors emphasize, “The ones you do partake in can be enjoyed guilt-free, knowing that your studies are still on track.”

2. Prepare Your Family

South University instructors recommend setting ground rules: “tell your family about the work you need to get done and let them know what you need.” Being honest about how much time you have, and what you think you can contribute to – while also emphasizing how important your high school equivalency is – can help your family understand why you need to spend a little time studying after dinner, or why you can’t play pick-up football like you usually do.

Sometimes it’s difficult to prepare your family for a change in a family tradition. Taking this time to explain will not only give your family members the opportunity to encourage and help you in whatever way they can. It will also prevent any interruptions to your schedule that get you off track.

3. Don’t Skimp on Sleep

If you’re a Black Friday shopper, this tip can be especially difficult.

However, one of the best things you can do is get enough sleep over the holidays. If you can, keep your schedule as close to normal as possible. This will help you stay on top of your studies and transition easily back into your work or school schedule after the Thanksgiving holiday is over.

According to the College Planning Group, part of maintaining your usual schedule should also be eating healthy. Your diet can significantly impact your sleeping cycle; for example, if you eat too much turkey on Thanksgiving and spend most of the afternoon napping, you might have trouble getting to sleep later that night. Thatthrows off your sleep cycle – and your study schedule. So even though it can be a struggle with family in from out of town and midnight shopping to do, “find ways to eat healthy and get sleep when you can.”

4. Take Time to Enjoy

Kelsey Sheehy of US News writes, “Don’t pretend like this is any other time of year.” The most important study tip for the holidays is to actually enjoy the holidays. Thanksgiving is a wonderful celebration, and you should feel as though you have time to give thanks with your family and friends.

If there are traditions you look forward to all year-round, those should be non-negotiable when you sit down to make your schedule. Or if you can’t bring yourself to study on Thanksgiving, be sure you manage the Wednesday before and the Friday after.

Remind yourself that this is a celebration of thanks and generosity. Use the excitement of the holiday to stay motivated and focused.

Happy Thanksgiving!

- The TASC Test Team


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