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Graphs with Holes - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

Rational functions have points where they are undefined, which introduces us to graphing holes in the function. Graphing holes involves being able to find these points. A rational function is a quotient of two functions, and if the denominator of this quotient has zeros, the rational function is undefined at that point. Graphing holes means showing what input values makes this denominator function zero.

We're talking about rational functions and I have one here that usually when we're analyzing the graph of a rational function we'd like to have the numerator and denominator factored, so let me do this just to figure out what's going on with this rational function.
I would factor the numerator starting with an x in each of these factors and I want to look for,80* let's say factors of 48 that are going to give me 14, so 6 and 8 and if I use -6 and -8 I'll get this exactly. I'll get -6 times -8 is 48 and -6x-8x is -14x and the denominator I can just pull a -2 out and that gives me -6+x, x-6, so this is interesting I've got the same factor in the numerator and denominator and I want to I want to know, is it okay just to cancel? Right, is this just the same as negative one half x-8 so we write equal question mark there.
Let's analyze the behavior of this function and the function before the canceling just to see what differences is differences are and the similarities are. I'll evaluate some some points let's try 2, when I plug in 2 I get 2-8, -6, 2-6, -4, -2, and then x-6 is again -4. This cancels and I get 3. If I plug 2 into this guy, 2-8 is -6 times negative a half also 3.
What about 4? I plug in 4 I get 4-88, -4, 4-6, -2, -2 and again the x-6 is -2, these cancel and I get 2. If I plug 4 in over here 4-8 is -4 times negative one half also 2.
If I plug in 6 this is going to be undefined. If I plug 6 in here I get 6-8, -2 times negative one half, 1. This is the key difference between the two functions because it seems like they have the same values everywhere else but in x=6 this one's undefined and that one is not. So the way to make these two equal to one another is say negative one half times the quantity x minus 8 for x not equal to 6.
Saying this, giving this domain restriction means that this function will now have the same domain as this function and now they're equal now you can write an equal sign here. The way to interpret this function is that its graph is going to be a line like right you know that the graph of this is going to be a line but there'll be a hole at x=6 that's what happens here and that's going to be really important when graphing rational functions that have these common factors between the numerator and denominator.

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