document.write( "Question 1196261: Choose the correct answer.
\n" ); document.write( "For real valued x, the function f(x) = (x-a)(x-b)(x-c) will assume all real values provided if the following condition is satisfied:
\n" ); document.write( "(A) a < b < c
\n" ); document.write( "(B) a > b > c
\n" ); document.write( "(C) a > c > b
\n" ); document.write( "(D) a < b < c
\n" ); document.write( "(E) b > a > c \n" ); document.write( "
Algebra.Com's Answer #829023 by math_tutor2020(3817)\"\" \"About 
You can put this solution on YOUR website!

\n" ); document.write( "It appears to be asking what values of a,b,c will make f(x) have a range of the set of all real numbers.\r
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\n" ); document.write( "\n" ); document.write( "If the function is \"f%28x%29+=+%28x-a%29%28x-b%29%28x-c%29\", then it doesn't matter what a,b,c are. The range of this is the set of all real numbers. Why? Because we have a cubic. All cubics can produce any real number output for some input(s).\r
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\n" ); document.write( "\n" ); document.write( "If instead the function is \"f%28x%29+=+%28%28x-a%29%28x-b%29%29%2F%28x-c%29\", then check out these links to see how to do a formal proof
\n" ); document.write( "https://www.youtube.com/watch?v=YDukXrhjpC0
\n" ); document.write( "and
\n" ); document.write( "https://byjus.com/question-answer/for-real-x-the-function-frac-x-a-x-b-x-c-will-assume-all-5/\r
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\n" ); document.write( "\n" ); document.write( "The conclusion is that c is between 'a' and b
\n" ); document.write( "So either a < c < b or b < c < a
\n" ); document.write( "Or a > c > b or b > c > a
\n" ); document.write( "All that matters is that c is in the middle. \r
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\n" ); document.write( "\n" ); document.write( "Answer: a > c > b\r
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\n" ); document.write( "\n" ); document.write( "Here's another approach you could take using numeric examples\r
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\n" ); document.write( "\n" ); document.write( "Let's say a = 1, b = 2, c = 3 which fits the a < b < c scenario
\n" ); document.write( "Then graph out \"f%28x%29+=+%28%28x-a%29%28x-b%29%29%2F%28x-c%29\", i.e. \"f%28x%29+=+%28%28x-1%29%28x-2%29%29%2F%28x-3%29\" in this case, and you would get this
\n" ); document.write( "https://www.desmos.com/calculator/umoamx9mlz
\n" ); document.write( "Desmos is a free graphing calculator.
\n" ); document.write( "Notice the gap of y outputs that cannot be reached. Eg: y = 2 is in that gap
\n" ); document.write( "The range of this function is not the set of all real numbers.
\n" ); document.write( "So a < b < c is clearly not the answer. \r
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\n" ); document.write( "\n" ); document.write( "Now try something like a = 3, b = 2, c = 1 which fits the a > b > c scenario
\n" ); document.write( "So we'll be graphing \"f%28x%29+=+%28%28x-3%29%28x-2%29%29%2F%28x-1%29\" to get this
\n" ); document.write( "https://www.desmos.com/calculator/frzlcau2hc
\n" ); document.write( "Once again we have a gap of y values we can't reach. Eg: we can't get y = -2 as an output
\n" ); document.write( "So we can rule out the case of a > b > c\r
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\n" ); document.write( "\n" ); document.write( "This trial and error process is continued until you find that a > c > b is the answer
\n" ); document.write( "Let's say a = 3, b = 1, c = 2
\n" ); document.write( "So we want to graph \"f%28x%29+=+%28%28x-3%29%28x-1%29%29%2F%28x-2%29\" to get this
\n" ); document.write( "https://www.desmos.com/calculator/puw4wq8rh2
\n" ); document.write( "There aren't any gaps for the y outputs now.
\n" ); document.write( "Any output is possible.
\n" ); document.write( "Therefore, the range of this is the set of all real numbers.
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