Functions - Domain, Range, Vertical Line Test

Functions may seem like a totally foreign and incomprehensible concept at first, but they're not really the monsters that many students think they are. The notation may seem a little weird from what we're used to, making them look difficult, but when you understand the concepts of what is being presented, you will see that they're not really doing anything that doesn't make sense.

To give a very simple definition, a function is basically just a way of relating two variables... although their notation makes it look otherwise. But we've already seen something like this before! If we take a regular X-Y graph and draw a straight line on it, the equation of the line is the way of relating the x variable with the y variable. However, the standard way of writing a function is to write f(x) instead of y. We say it like f of x. So for example, taking a line, we can do the following:

y = 3x + 5
f(x) = 3x + 5

The function notation says that "the whole expression with x on the right" evaluates to some number (the equivalent of y). The math is still the same, no tricks with what you can do with the expression. It's just a different way of describing things. But we also now have a way of saying "tell me what that thing evaluates to when x=10" and we can write that out like this:

f(10) = 3(10) + 5 = 30 + 5 = 35
f(10) = 35

Again, it's essentially the same thing as saying 'solve for y' from our graph equations, but you can't say to solve y = 3x + 5 at a certain point for x, without actually writing that instruction out in words. With function notation, you can say to solve the expression when x=10, 50, or 25,000, just by writing f(10), f(50), or f(25,000).

There are a few other terms you will likely see when dealing with functions. Domain refers to all the values of x that are valid for the expression. Range refers to all the values of y in the expression. Domains and ranges could be infinite, as in the case of a straight line that goes forever in either direction, or they could have defined numbers as in line segments. The notation for domain and range sets is like [x₁, x₂] or [y₁, y₂], where those numbers represent the two extremes of the domain (furthest left and right) or range (highest and lowest), and the square brackets indicate that those numbers are included in the range. (If you had an 'open' point on a line, this indicates that that exact point is excluded from the line, and if this occurred as an endpoint, you would write (x1, x2) to denote that the number is not included in the set. Also, you could have on the line a closed and an open point, in which case you'd have something like (x1, x2] or vice versa.)

The domain (the x-values) for line segment A is [-8, 1] and its range (y-values) is [5, 8]. In this case the domain and range extremes are the actual end points of the line, but pay attention to the particular question as this is not always the case. Remember, this notation is defining the domain and range (all the values that x or y could take), not the end points of the line.

Similarly for line segment B, the domain is [2, 7] and the range is [-3, 6].

Line C demonstrates a very important concept about functions. Line C is not a function. We should expand our definition of a function to say that it is a way of relating two variables, where each number in the domain corresponds to ONLY ONE number in the range. That means, that for every x, there can only be one y. To test this on a graph, you can use the vertical line test to prove a line is a function or not. If you draw a vertical line through your function at any value for x, the vertical line will pass through the function line ONLY ONCE. If it passes through it more than once, then you are not looking at a graph of a function. Therefore, looking back to line c, you can tell that a vertical line would pass through the line twice at some positions, and so this proves it is not a function of x. Similarly, the circle in D does not pass the vertical line test, and so is not a function either.

These concepts may seem different, but you will become accustomed to them. Just remember our definition of what a function actually is, and what the notation is saying, and you will be fine. When you begin to grasp these concepts and advance to higher levels, you will get to see some really cool stuff with functions. (sneak peek.... f(x) = x⁶ + x⁵ + x⁴ + x³ + x² + x + 1.... it's still a function!)

If any of that isn't clear and you'd like something more explained or clarified, please don't hesitate to leave a comment for me! I'm always glad to help!

Graphing - Standard Form of the Equation

Just a short explanation for what is meant by "standard form" of the equation of the line. We have been looking at line equations in the form of y=mx+b. However, you may be asked to express this in standard form, or as a standard form equation.  Graphing standard form equations will give you the exact same line as graphing something expressed as y=mx+b... standard form is just a different way of displaying the equation.  (Please hit the Like button and/or the +1 button if this post is helpful for you!)

The general notation for a standard form equation is Ax + By = C, where A, B, and C are coefficients, and the x and y are the same variables we've been looking at but in a different position from what we recognize.

To express in standard form, you simply just rearrange the y = mx + b form such that you have x and y on the same side, equal to a number. Let's look at some examples:

Given that y = 3 x+ 5, standard form of this is 3x - y = (-5).

Given y = (1/2)x -15, standard form of this is (1/2)x - y = 15... also, if you don't want to have any fractions in your answer, you can multiply everything by the number in the denominator, such that we now get x - 2y = 30. Both expressions mean the same thing and will produce the same line. (In fact, convince yourself that no matter what you do to the equation, so long as you do it to both sides, the line is the same. eg. Multiply it all by 100, you get 100x-200y=30000... looks different, but it's not! Reduce it down and see for yourself!)

For graphing standard form equations, you still might want to go from standard form to the mx+b form, for which you may need to do a bit more math, but it's still quite straight forward.

Given 5x - 15y = 10, you just have to rearrange things to get y by itself on one side:
(-15y) = (-5x) + 10
y = (1/3)x - (2/3)...
and then you can see it is a line with slope 1/3 and y-intercept (-2/3).

Both types of equations mean the same thing. They are just expressed differently, and y=mx+b gives immediate information about the line without having to do a lot of work. However, you should be able to use both forms interchangeably.  Convince yourself that graphing standard form equations will give you the same line as graphing y=mx+b equations.  They just look different because the numbers are rearranged.  This should be obvious because if you start with a standard form equation, and convert it to y=mx+b and graph it, you have only rearranged things not added or removed anything.  You do not have a new line.

Also, from these equations, you should be able to tell that whenever you have an equation with 2 variables (x and y), and there aren't any exponents on either term, then you are dealing with a straight line. So while an equation in standard form may not immediately look like a straight line equation to you until it looks like y = mx + b, because it has an x and a y in it (without an exponent... exponents make the graph do cool things later), it is automatically a straight line.

(check this post for some additional pointers)

If you liked my post, you can help me spread the word by tweeting about it!  Also, please remember to click the Like button or +1 if it helped!