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Hi, I’m Rob. Welcome to Math Antics!
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In this algebra basics lesson, we’re gonna learn about functions.
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Outside of the realm of math, the word “function” simply refers to what something does.
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But in math, the word “function” has a more specific meaning.
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In math, a function is basically something that relates or connects one “set” to another “set” in a particular way.
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A set is just a group or collection of things.
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Often it’s a collections of numbers, but it doesn’t have to be.
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A set could be a collection of other things like letters, names, or just about anything.
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Sets are sometimes shown visually like this,
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but more often you’ll see sets written using a common math notation
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where some or all of the members of the set are put
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inside curly brackets with commas between them like this.
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A set can have a finite or an infinite number of elements.
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For example, a set containing all the letters of the alphabet has only 26 elements,
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while the set of all integers has an infinite number of elements.
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Okay, so a set is just a collection of things, and a function relates one set to another.
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But how exactly does it do that?
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Well, to understand how functions work it will help if we start by naming the two sets the input set and the output set.
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A function is something that takes each value from an input set and relates it (or maps it) to a value in an output set.
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And you’ll often hear these input and output sets referred to by special math names.
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The input set is usually called “The Domain”
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and the output set is usually called “The Range”.
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And it’s really common to see some or all of a function’s inputs and outputs listed in what we call a “function table”.
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A function table normally has two columns:
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one on the left for the input values
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and one on the right for the corresponding output values.
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The function itself is often written above the function table
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and in the form of some sort of mathematical rule or procedure.
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For example, let’s say that the input set of a function is a list of common polygon names like
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{triangle, square, pentagon, hexagon and octagon}
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The function itself could be a simple rule that says, “Output the number of sides.”
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That means, if we input “triangle” into the function, the output will be 3.
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And if we input “square” the output will be 4.
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If we input “pentagon”, the output will be 5, and so on…
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So this function simply relates the name of a polygon to its number of sides.
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That’s cool… but most of the functions that you’ll encounter in Algebra will be a little more abstract than that.
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They’ll usually just relate one variable to another variable in the form of an equation …like this one: y = 2x
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In this equation, if we treat ‘x’ as the set of numbers that we can input (the domain),
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and ‘y’ as the set of numbers that we get as outputs (the range),
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what we have is a very simple algebraic function.
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And just like the polygon example, we can make a function table to show some of the possible input-output combinations.
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For this function, we could choose any number at all for the value of ‘x’,
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but to keep things simple, let’s just try inputting 1, 2, and 3 as values of ‘x’
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and see what outputs we get for our table.
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If we input the value 1 (in other words, if we substitute the value 1 for the ‘x’ in our equation)
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then we get y = 2 × 1 which simplifies to y = 2.
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And since ‘y’ is our output variable, we put a 2 in the output column.
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Next, if we input the value 2 into our function,
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we get y = 2 × 2, which means y = 4. So the output value is 4.
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And last, if we input the value 3 into our function,
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we get y = 2 × 3, which means y = 6. So the output value is 6.
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For each input value, the output value is twice as big.
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Which is what we would expect because the original equation says that 'y' (the output) is equal to 2 times ‘x’ (the input)
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Okay, so we’ve seen some examples of functions that relate inputs to outputs,
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but there’s an important limitation about functions that we need to know.
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To understand what that limitation is, let’s try to make a function table for the equation ‘y squared’ equals ‘x’.
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Again, the ‘x’ variable in this equation will be our set of inputs and the ‘y’ variable will be our set of outputs.
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Since ‘y’ is our output variable, it will help if we first solve this equation for ‘y’
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and we do that by taking the square root of both sides.
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But because of negative numbers, we need to take both the positive and negative root of ‘x’
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since there are two possible solutions to our equation.
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But, won’t that mess up our function table?
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If we input an ‘x’ value of 4, the positive (or principal) root would be 2,
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but we also have the negative root as a solution.
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If x = 4, then y = 2 and y = -2 are BOTH possible solutions to the equation ‘y squared’ equals ‘x’.
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So in this case, for each value of ‘x’ that we input into the equation, we’ll get TWO values of ‘y’ as outputs.
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Can a function do that?
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Upon review, the equation gave two outputs for a single input,
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therefore it’s ruled not a function.
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You see, functions aren’t allowed to have what we call “one-to-many” relations,
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where one particular input value could result in many different output values.
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“One-to-many” relations certainly do exist as we can see from this example, but we don’t call them functions.
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For something to be called a function, it has to produce only one output value for each input value.
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So a function doesn’t just relates a set of inputs to a set of outputs.
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A function relates a member of an input set to exactly one member of an output set.
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The equation y = 2x qualifies as function because no matter what number you put in,
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you’ll always get just one number as an output.
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But the equation ‘y squared’ equals ‘x’ does not qualify as a function
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because a single input can produce more than one output.
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Let’s look at another simple algebraic equation to see if it’s a function:
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Again, the ‘x’ values will be inputs (the domain) and the ‘y’ values will be the outputs (the range).
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Let’s quickly generate a function table for a few possible input values, like the integers -3 through +3.
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If you watched our last video about graphing on the coordinate plane,
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you may notice that each row of this function table is basically just an ordered pair.
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It’s an ‘x’ value followed by a ‘y’ value.
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We could even re-write all the inputs and outputs in ordered pair form if we wanted to.
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And that means, you can also GRAPH all of these pairs of inputs and outputs on the coordinate plane.
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You can GRAPH a function!
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Here are the points from our function table plotted on the coordinate plane,
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and here’s the resulting graph we get if we connect those points.
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It forms a straight line and it’s an example of what is called a “linear function”.
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In Algebra, there are lots of different kinds of functions that have interesting graphs:
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quadratic functions,
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These graphs may look like just a bunch of squiggly lines, but they’re all functions.
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And we tell they’re functions just by looking at their graphs because they all pass the “Vertical Line Test”.
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Remember how functions aren’t allowed to have more than one output value for a particular input value?
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Well, the Vertical Line Test helps us see if a graph has any of those
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one-to-many relations that would disqualify it as a function.
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Here’s how it works…
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Imagine that a vertical line is drawn on the same
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coordinate plane as the graph that you want to test.
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Then, imagine moving that vertical line left and right
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across the domain, paying close attention to the point
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where the vertical line intersects with the graph.
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If that vertical line only intersects the graph
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at exactly one point for every possible value of ‘x’ in the domain,
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then that means there’s only one output value for each input value.
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There’s only one ‘y’ value for each ‘x’ value
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so the graph qualifies as a function.
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Okay, so all of these graphs pass the Vertical Line Test and are functions.
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But what’s an example of a graph that doesn’t pass the Vertical Line Test?
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Well here’s one. It’s the graph of our equation ‘y squared’ equals ‘x’.
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The domain of this equation doesn’t include any negative input values,
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so there are some places where our vertical line wouldn’t interest the graph at all.
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And there’s one place where the vertical line would intersect the graph at just one point,
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But, as we move to the right on the ‘x’ axis,
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you can see that our vertical line is now intersecting the curve in TWO places.
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That means this equation is giving us two possible outputs for some of its inputs,
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which means that it’s not considered a function.
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Okay… now before we wrap up,
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we need to talk briefly about some common function notation that can be pretty confusing
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the first time you see it in math books.
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So far, we’ve been writing functions like this: y = 2x and y = x + 1
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But you’ll often see these same exact functions written like this instead.
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Why did the variable ‘y’ get replaced by that ‘f’ parentheses ‘x’ thingy?
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And what does that even mean?
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Well, it turns out that a really common way to represent a function is this…
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This notation simply means that a function (named ‘f’) takes an input value (named ‘x’) and gives an output value (named ‘y’)
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And you say it like this:
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A function of ‘x’ equals ‘y’
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or ‘f’ of ‘x’ equals ‘y’ for short.
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The problem with this notation is that you could easily misinterpret it
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as a variable ‘f’ being multiplied implicitly by a variable ‘x’ to give an answer of ‘y’.
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But that’s NOT what this means.
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In this case, ’f’ is not the name of a variable and it’s not being multiplied.
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Instead, ’f’ is the name of the function.
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It would be a lot more clear if mathematicians just used the entire word “function” as the name
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and then used the names “input” and “output” instead of ‘x’ and ‘y’.
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These two notations mean exactly the same thing.
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But the first one uses an abbreviation for the function name and standard variable names for the input and output.
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These are the most common names, but you could use others if you wanted to.
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Okay, so that’s the basic notation, but how did the equation get changed to f(x) instead of ‘y’?
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Well, it comes from the idea that if two things are equal in math, you can substitute one thing for the other.
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Since we’ve agreed on this general notation for a function, f(x) = y,
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that means you can use f(x) or ‘y’ interchangeable.
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Either one can represent the output set of a function.
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But if they’re interchangeable, why would you use the more complicated f(x) when you could just use ‘y’ instead?
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Well, using f(x) highlights the fact that you’re dealing with a function
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with a specific input variable and not just an equation.
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And… it gives us a handy notation for evaluating functions for specific values.
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For example, you could start off by saying, let the function f(x) = 3x + 2.
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Then you could then ask someone to evaluate the function for the input value 4 by saying what is f(4).
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That means you’ll substitute a 4 in place of any ‘x’s that are in the function.
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For this function, that would mean f(4) = 14.
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And you could do this for other values too.
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Alright, so that’s what functions are in math.
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They’re things that relate an input value to exactly one output value.
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And the set of all input values is called the domain
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while the set of output values is usually called the range.
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In algebra, functions typically come in the form of equations that can be graphed
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on the coordinate plane by treating the input and output values as ordered pairs.
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Of course, there’s a LOT more to learn about functions,
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but this basic introduction should help you get started working with them in Algebra.
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Don’t forget to practice using what you’ve learned in this video by doing some exercises.
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As always, thanks for watching Math Antics and I’ll see ya next time.
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Learn more at www.mathantics.com
