document.write( "Question 135067: What is the difference between domain and range? Describe a real-life situation that could be modeled by a function.\r
\n" ); document.write( "\n" ); document.write( "Can someone explain this so I can understand it? \n" ); document.write( "

Algebra.Com's Answer #98906 by solver91311(24713) $\"\"$ $\"About$

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Think of a function as a machine. It looks like a little box with a slot in the top and little tray on the side. You put numbers into the slot and different (usually) numbers come out in the little tray. I say usually because if the function were f(x) = x, then you would get the same number out that you put in.\r
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\n" ); document.write( "\n" ); document.write( "Now, if there are certain numbers that would cause the machine to malfunction, usually either because you would be trying to divide by zero or put a negative number under a radical, then you would have to put a warning sign on the box telling you not to put certain numbers into the slot.\r
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\n" ); document.write( "\n" ); document.write( "Example: $\"f%28x%29=x%2F%28x-2%29\"$ . Clearly 2 would make the denominator go to zero, and your little machine would either melt down or blow up. So you would need a sign on it that says \"WARNING: Do Not Insert the Number 2\"\r
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\n" ); document.write( "\n" ); document.write( "The domain of a function is the set of numbers that CAN go in the slot. For some functions, this set is all real numbers, as in $\"f%28x%29=x%5E2-5x%2B4\"$ . For other functions, you may have to exclude certain discrete numbers, as in $\"f%28x%29=4%2F%28x%5E2-2%29\"$ where either $\"sqrt%282%29\"$ or $\"-sqrt%282%29\"$ would damage the machine. Still other functions might require you to exclude entire ranges of numbers in order to be defined over the real numbers, example: $\"f%28x%29=sqrt%28x-2%29\"$ . Here, x must be greater than or equal to 2 so that the value under the radical is never negative.\r
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\n" ); document.write( "\n" ); document.write( "The range of a function is the set of numbers that comes out in the little tray. If you took your machine and put in every possible number in the domain, the range would be the set of numbers that came out the other end.\r
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\n" ); document.write( "\n" ); document.write( "Let's look at some examples:\r
\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=x%2F%28x-2%29\"$ : When x gets close to 2, the function gets very large. Once you get to calculus, you will prove that $\"lim%28+x-%3E2%2C+%28x%2F%28x-2%29%29+%29=infinity+\"$ or $\"-infinity\"$ depending on whether you approach 2 from the right or the left. Any x on the interval [0,2) gives a negative result, everything else (except 2) gives a positive result. That covers the entire set of real numbers. So the range is all reals.\r
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\n" ); document.write( "\n" ); document.write( " $\"f%28x%29=x%5E2-5x%2B4\"$ : This is a parabola that is concave up, so the vertex is a minimum. The vertex is at ( $\"5%2F2\"$ , $\"%28-9%29%2F4\"$ ), so the range is [ $\"%28-9%2F4%29\"$ , $\"infinity\"$ )\r
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\n" ); document.write( "\n" ); document.write( "Hope that helps.\r
\n" ); document.write( "\n" ); document.write( "John \n" ); document.write( "

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