document.write( "Question 139478This question is from textbook Pre- Algebra
\n" ); document.write( ": I don't really understand functions. \n" ); document.write( "
Algebra.Com's Answer #102861 by solver91311(24713)\"\" \"About 
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A function is a little black box, it has a slot on one end and a tray on the other. You put values of x into the slot, and get values of y out in the tray. Only one important rule: For any given value of x you put in, you can only get one value of y out, otherwise it is just a relation but not a function. That doesn't mean that you can't have two or more values of x that give you the same y value though.\r
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\n" ); document.write( "\n" ); document.write( "Some functions have a warning sign on them, like \"Only positive numbers allowed\" or \"You can insert any number except 3\" This is the Domain of the function.\r
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\n" ); document.write( "\n" ); document.write( "The set of all possible numbers that could come out into the tray is the Range of the function.\r
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\n" ); document.write( "\n" ); document.write( "Function notation:\r
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\n" ); document.write( "\n" ); document.write( "f (or g or h or whatever) is the function. f(x) is the value of the function when x is inserted into the slot. f(2) is the value of the function when 2 is inserted into the slot, and so on.\r
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\n" ); document.write( "\n" ); document.write( "For example, let's define a function f as f(x) = x + 2. Then f(2) would be 2 + 2 = 4, f(a) would be a + 2, etc.\r
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\n" ); document.write( "\n" ); document.write( "Domain example:
\n" ); document.write( "Let g be a function defined by \"g%28x%29=%28x%5E2-5x%2B4%29%2F%28x-1%29\". We all know that we can't have a zero denominator in a rational expression, so the little black box for this function is going to have a warning sign that says: \"Insert any number except 1\" In the arcane jargon of mathematics, one would say that \"the domain of g is the set of all real numbers x such that x is not equal to 1\"\r
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\n" ); document.write( "\n" ); document.write( "Another domain example:
\n" ); document.write( "Let h be a function defined by \"h%28x%29=sqrt%28x%29\". Here we need to decide how we want to define the function. If we want the function defined over the real numbers, then we have to make sure that the value under the radical is never less than zero, so: the domain of h is the set of all real numbers such that x is greater than or equal to 0. On the other hand, if we can define the numbers over the complex numbers, then any value can be inserted.\r
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\n" ); document.write( "\n" ); document.write( "Range example:
\n" ); document.write( "Let f be a function defined by \"f%28x%29=x%5E2%2Bx-6\". We know that this is a parabola that opens upward with vertex at (\"-1%2F2\",\"-25%2F4\") (proof left to the student). So the smallest value the function can be is \"-25%2F4\", but there is no restriction on the maximum value. Hence, the range is the set of all y such that y is a real number and y is greater than or equal to \"-25%2F4\"\r
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\n" ); document.write( "\n" ); document.write( "Another range example:
\n" ); document.write( "Let f be a function defined by \"f%28x%29=sqrt%28x%2B3%29\". The radical implies that we are dealing with the positive square root. It cannot mean both the positive AND negative square root because then we wouldn't have a function (we can't have two values for every x value). So, this function has a range from -3 to infinity, or the set of all y such that y is a real number and y is greater than or equal to -3.\r
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\n" ); document.write( "\n" ); document.write( "Composite functions:
\n" ); document.write( "Sometimes you will see something like: \"f%28x%29=x%5E2-4\" and \"g%28x%29=1%2Fx\", what is (f°g)(x)? This is called the composite of f and g, and it simply means: \"f%28g%28x%29%29\". Remember \"f%28x%29\" means the value of f at x. So \"f%28g%28x%29%29\" just means the value of f at the value of g at x. That's a lot of words to mean take the expression for g(x) and substitute it for x in the expression for f(x).\r
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\n" ); document.write( "\n" ); document.write( "So: if \"f%28x%29=x%5E2-4\" and \"g%28x%29=1%2Fx\", (f°g)(x) = \"%281%2Fx%29%5E2-4\". On the other hand, (g°f)(x) would be \"1%2F%28x%5E2-4%29\".\r
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\n" ); document.write( "\n" ); document.write( "The one really important take away from all of this is: \"f%28x%29\" is read \"f of x\", but I want you to think \"\"f%28x%29\" is the value of the function f at x\"\r
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\n" ); document.write( "\n" ); document.write( "Hope that helps.
\n" ); document.write( "John\r
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