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One-to-One Functions: Definition, Examples, and Applications

Algebra: Master Linear Equations & Problem Solving

Basic Algebra

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Model and solve one-step linear equations: ax = b, x/a = b

One-step Linear Equations

One Step Linear Equations — Word Problem

One-step equations

One-step Equations LHS

One-step Equations RHS

Model And Solve One Step Linear Equations Divide – Word Problem

Model And Solve One Step Linear Equations Multiply – Word Problem

Solving two-step linear equations using addition and subtraction: ax + b = c

Two-step Linear Equations

Two-step Linear Equations LHS

Solve Linear Equations Using Addition and Subtraction – Word Problem

Two-step Linear Equations — Word Problem

Solving two-step linear equations using multiplication and division: x/a + b = c

Two-step Linear Equations RHS

Solving two-step linear equations using distributive property: a(x + b) = c

Solve Linear Equations Using Distributive Property – Word Problem

Patterns

Patterns — Word Problem

Evaluating algebraic expressions

Evaluating Algebraic Expressions

Solving one - step equations: x + a = b

1-step Equations

1-step Equations — Word Problem

Solving literal equations

Exponents

Using exponents to describe numbers

Expanded powers

Exponent rules

Single Power Expressions

Dividing powers

Multiplying powers

Order of operations with exponents

Using exponents to solve problems

Product rule of exponents

Exponents: Product rule (a^x)(a^y) = a^(x+y)

Exponents: Product rule (a^x)(a^y) = a^(x+y)

Exponents: Division rule (a^x / a^y) = a^(x-y)

Exponents: Division rule: a^x / a^y = a^(x-y)

Exponents: Power rule (a^x)^y = a^(x * y)

Exponents: Power rule: (a^x)^y = a^(xy)

Exponents: Power rule

Exponents: Negative exponents

Exponents: Zero exponent: a^0 = 1

Exponents: Zero exponent: a^0 = 1

Exponents: Rational exponents

Quotient rule of exponents

Power of a product rule 

Power of a quotient rule 

Power of a power rule 

Simplify the power

Power of a power rule

Negative exponent rule 

Combining the exponent rules

Scientific notation

Solving for exponents

Introduction to Exponents

Exponents — Express as a power of 10

Exponents — Word problem: express as a power

Exponents — Express as a power of two — Whole number

Radicals

Conversion between entire radicals and mixed radicals

Adding and subtracting radicals (Advanced)

Adding and subtracting radicals

Combining Like Radicals

Adding and Subtracting Radicals

Multiplying radicals (Advanced)

Multiplying Radicals

Multiplying radicals

Squares and square roots

Square Root of a Perfect Square

Estimating square roots

Operations with radicals

Operations with Radicals

Evaluating and simplifying radicals

Convert between radicals and rational exponents

Square and square roots

Cubic and cube roots

Cube root of a perfect cube

Converting radicals to mixed radicals

Converting radicals to entire radicals

Adding radicals

Subtracting radicals

Multiplying and dividing radicals

Dividing radicals

Rationalize the denominator 

Unit Conversion

Conversions between metric and imperial systems

Metric to Imperial Conversions

Imperial to Metric Conversions

Conversions Between Metric and Imperial Systems Speed KM to MI

Conversions Between Metric and Imperial Systems Speed  MI to M

Metric-Imperial Length Conversion Inch To Cm

Metric-Imperial Length Conversion MM/CM/M/KM To FT

Metric-Imperial Length Conversion MM/CM/M/KM To MI

Metric systems

Add and Subtract Mixed Units

Metric Length Conversion

Add or Subtract Different Metric Units Convert to CM

Add or Subtract Different Metric Units Convert to M

Add or Subtract Different Metric Units Convert to MM

Imperial systems

Imperial Length Conversion

Conversions involve squares and cubic

Imperial to Metric Conversions — Squares and Cubes

Imperial Conversion — Squares and Cubes

Metric to Imperial Conversion — Squares and Cubes

Conversions involving squares and cubic

Metric Area Conversion FT to CM

Metric Area Conversion Foot to Inch

Metric Area Conversion M/CM to Inch

Linear Equations

Introduction to linear equations

Introduction to nonlinear equations

Special case of linear equations: Horizontal lines

Special case of linear equations: Vertical lines

Parallel line equation

Perpendicular line equation

Combination of both parallel and perpendicular line equations

Applications of linear equations

Representing patterns in linear relations

Represent Patterns in Linear Relations Equation Decimal

Represent Patterns in Linear Relations Equation Integer Start From 0

Represent Patterns in Linear Relations Equation Integer Start From 1

Represent Patterns in Linear Relations Equation Negative Decimal

Identify the Linear Equation

Reading linear relation graphs

Solving linear equations by graphing

Solving linear equations using multiplication and division

Solving two-step linear equations: ax + b = c, x/a + b = c

Solving two-step linear equations: ax + b = c, x/a + b = c

Solving linear equations using distributive property: a(x + b) = c

Solving linear equations using distributive property: a(x + b) = c

Solving linear equations with variables on both sides

Understanding graphs of linear relationships

Linear relations

Understanding tables of values of linear relationships

Linear relation

Applications of linear relationships

Graphing linear relations

Identifying proportional relationships

Linear Functions

Distance formula: \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Distance formula: <katex>d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}</katex>

Distance formula

Midpoint formula: \(M = ( \frac{x_1+x_2}2  ,\frac{y_1+y_2}2)\)

Midpoint formula

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)</katex>

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)</katex>

Slope equation: \(m = \frac{y_2-y_1}{x_2- x_1}\)

Slope equation: <katex>m = \frac{y_2-y_1}{x_2- x_1}</katex>

Slope intercept form: y = mx + b

General form: Ax + By + C = 0

Point-slope form: \(y - y_1 = m (x - x_1)\)

Graphing linear functions using table of values

Graphing linear functions using x- and y-intercepts

Graphing linear functions using various forms

Graphing from slope-intercept form y=mx+b

Graphing linear functions using a single point and slope

math

<h2>What is Algebra?</h2>
Algebra is the study of mathematical symbols and the rules surrounding the manipulation of these symbols. The symbols are used to represent quantities and numbers within equations and formulae.
We use symbols in algebra because of the unknown values, known as variables. If there is no known value (number) like 10 or 5, we place a letter or symbol (usually X or a Y ) to represent that the value is unknown. In contrast to this, there are elements of algebraic equations called "constants". These get their name because the value remains the same throughout. For example, the number 23 will always represent 23 units and never another value.
As an Example: X + 23 = 28
In the basic equation above, the variable (x) is the number required to make this sum work. All we need to do is to add another number to 23 (constant) so that we can get 28 (constant). Adding 5 to 23, gives you 28, so now we know that the value of x is 5.
Algebra is considered a unifying concept of almost all elements of math and as such, it stretches far beyond elementary equations and into the study of things like, radicals, integers, linear systems, exponent rules, quadratic equations, the Pythagorean Theorem and much more.

<h2>Who Invented Algebra?</h2>
The roots of algebra can be traced back to the ancient babylonian era. The babylonians had advanced systems of math that allowed them to calculate things in an algorithmic fashion. For the longest time, It was thought that the "father of algebra" was Diophantus, an alexandrian greek mathematician. This has recently been challenged though, with historians and mathematicians suggesting that the title should be with al-Khwarizmi, the founder of the discipline of al-jabr. It&apos;s worth noting that the term "al-jabr", meaning reunion of broken parts, is where we get the word algebra from.

<h2>How to Learn Algebra?</h2>
With algebra, like anything in math, the best way to learn is with repetition. Practice makes perfect, so make time outside of school/college hours to review your class notes. If you aren&apos;t taking notes in class, you really should be. These notes will not only help you retain the information but they can also be used to help you identify any areas of weakness that you have in relation to algebra. It&apos;s important that you know where your strengths and weaknesses are so that you can build effective revision strategies ahead of your upcoming exams. Highlight key areas for improvement and get yourself into a good study habit.
When it comes time to revise, use StudyPug&apos;s vast collection of algebra lessons to help you. With step-by-step examples for even the toughest algebra problems, we offer easy practice solutions for high school algebra content through to more complex college level algebra. With your very own algebra tutor, you can select the lessons that are relevant to you, making for a tailored study session that works to strengthen the weaker areas of your knowledge.
If you&apos;re looking for algebra homework help, we&apos;ve got you covered. Additionally, we&apos;ll also assist you with your algebra exam prep, covering all the content you&apos;d expect to find on the actual papers. As mentioned above, our algebra tutoring sessions cover all skill levels, so if you&apos;re looking for an introduction to algebra, we can get you started. Our content will walk you through each aspect of algebra and when you&apos;re ready, you can progress to the next step. In no time at all, you&apos;ll be solving equations, understanding algebra formulas and so much more! This is because our content is delivered via video tutorials that can be paused, rewound, and fast-forwarded, ensuring that you can learn at your own pace and will never get left behind.
Many of our students find the video format is a lot more palatable and easier to follow than traditional textbooks. That&apos;s because our online content avoids overly complex terminology and presents it in a way that's much easier to understand. The videos are presented by knowledgeable math teachers who have worked to ensure that their video content covers the same content found in class and on exams.

<h2>What Grade Level is Algebra?</h2>
Algebra is one of the foundations of mathematics and as such, it's taught in one form or another at all grade levels. It is taught to young students once they have learnt how to find the pattern in numbers, and with each passing grade, the students continue to build their understanding of the more complex elements eventually learning abstract algebra at college and university.
Below we have listed the common algebra focused courses and at what grade you can expect to study them
<ul type="disc">
<li>Pre-algebra: Usually Studied in 5th or 6th Grade</li>
<li>Basic Algebra: Usually Studied in 7th or 8th Grade</li>
<li>Algebra I: Usually Studied in 9th Grade</li>
<li>Algebra II: Usually Studied in 11th Grade</li>
<li>Intermediate Algebra: Taken Before Any University Level Math, E.G. College Algebra or Calculus</li>
<li>College Algebra: First Year of College or University</li>
</ul>

<h2>How Can I Solve and Simplify Algebraic Equations Without a Calculator?</h2>
In order to solve algebraic equations without the use of a calculator, you&apos;re going to need to understand algebra instead of simply memorizing the formulae. Once you have a firm grasp on how the math works, you&apos;ll be able to tackle complex math problems without a calculator.
You should consider browsing our website for helpful videos that break down math into step-by-step examples so that you not only memorize them but actually understand them too. We also have useful tips on how to solve other problems with calculators, such as <a href="[[canon_topic.html|{"textbook_key":"algebra-help","chapter_key":"radicals","topic_key":"estimate-square-roots"}]]">estimating square roots</a>, <a href="[[canon_topic.html|{"textbook_key":"basic-math-help","chapter_key":"percents","topic_key":"percent-of-a-number"}]]">percent of a number</a>, and <a href="[[canon_topic.html|{"textbook_key":"algebra-help","chapter_key":"logarithm","topic_key":"evaluate-logarithms-without-a-calculator"}]]">evaluating logarithms</a>.
It's also worth noting that math hasn&apos;t changed and there was a time before calculators. Track down some older textbooks and you may find some useful tips for solving math problems with only a pen, some paper, and a little mental arithmetic.

https://dmn92m25mtw4z.cloudfront.net/img/textbook/textbook-algebra1.png

Understanding One-to-One Functions: Examples and Real-World Applications

What You'll Learn

Distinguish between functions, injective (one-to-one), and surjective (onto) functions

Identify one-to-one functions using the horizontal line test on graphs

Apply the vertical line test to determine if a relation is a function

Recognize that injective functions have at most one x-value for each y-value

Understand that surjective functions map to all y-values with no outputs left unmapped

What You'll Practice

Testing graphs using horizontal and vertical line tests

Analyzing mapping diagrams to classify function types

Identifying one-to-many vs. many-to-one relationships

Determining whether functions are injective, surjective, or neither

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

4 Video lessons

Practice exercises

Learning resources

Skills

One-to-One Functions

Injective Functions

Surjective Functions

Horizontal Line Test

Vertical Line Test

Function Mapping

Domain and Range