Algebra 2 Function Notation and Testing
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A Y in Disguise
Function Notation and Testing

 

 Up to now, we have seen a lot of equations that look like this:

 

 Y = (stuff with X's in it)

 
 Now we are going to give Y another name.
 It's new name is:

 F(X)

 
 This is pronounced "F of X"
 It's really just another name for Y.
 
 What it means is ...
 (just skip this part if you don't care what it means)
 Remember when we said that in these equations
 X could be pretty much whatever it wants to be
 and that Y depended on X.
 That made Y the dependent variable
 and X the independent variable.
 
 Another way to say that "depended on" thing is
 "is a function of"
 So:
 Y depends on X
 Y is a function of X
 Y is F of X
 Y is F(X)
 Y = F(X)
 It's sort of like the evolution of math ...
 
 OK, that's it for "What does F(X) mean?"
 You should come back now ...
 
 So instead of 

 Y = 3X2 + 2

 
 You could say ...

 F(X) = 3X2 + 2

 
 This name for Y is like a special club.
 Not just an old Y = (stuff) equation 
 can use this new name for Y.
 To join the club, the equation must pass a test.
 The test is called the vertical line test.
 Here's how it works.
 
 First draw the graph of the equation you're testing.
 We'll try Y = 1/3X 2 + 2.
 This is a "U" that has it's bottom at X = 0, Y = 2.
 

 

 
 Now here's the test.
 If we can draw a vertical line 
 (vertical means "goes straight up and down")
 that crosses the graph line MORE THAN ONCE
 the equation does not get to join the F(X) club.
 
 In this case, there is no vertical line that we can draw
 that crosses the equation line more than once.
 So Y = 1/3X 2 + 2 gets to use the new name for Y.
 

 Y = 1/3X2 + 2 

 
 OK fine, but we could have some really complicated equation
 that could take forever to graph.
 Even then, there could be some little place out at X = ten million
 that we might miss.
 That point might be the one and only place a vertical line
 hits the equation more than once.
 How about some rules that say what kinds of equations
 meet or fail the test?
 
 No problem ...
 
 When you have equations with stuff like:
 

 Y = 4X6 - 3X4 + 2X3 - X2 + 5

 
 They will pass the test.
 They key is, we just have Y on the left (no exponents on Y)
 and just X (exponents OK) and numbers on the right.
 
 If we have 2Y = stuff with X's in it 
 we can divide both sides by a number (2) and get Y = stuff with X's in it
 so it passes ...
 
 If we have Y + 3 = stuff with X's in it,
 we can subtract 3 from both sides to make it Y = stuff with X's
 so it passes.
 
 BUT ...
 
 If we have:
 Y2 = stuff with X's in it

 IT FAILS

 
 If we have:
 |Y| = stuff with X's in it

 IT FAILS

 
 If we have:
 Y =  (stuff with X's in it)

 IT FAILS

 
 If the equation gets to use F(X)
 It also gets to call itself a function.
 

   copyright 2005 Bruce Kirkpatrick

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