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You can't handle the Shmoop.

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If  f of x equals x squared minus 100 over x squared plus 10x,

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and f of x minus 5 equals 1/3, then x must be which of the following?

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And here are the potential answers…

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All right, so we’re trying to find out what x should be given two functions.

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Well, the first function we’re given, x squared minus 100 over x squared plus 10x, can be factored.

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X squared minus 100 looks a heck of a lot like the difference of two squares….

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so the numerator becomes (x + 10) times (x - 10). And the denominator is x squared plus 10x…

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which becomes x times (x + 10). There’s an x + 10 now in both the numerator

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and denominator, so we can cancel out the x +10 in each place.

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Our new equation looks like this: f of x equals x minus 10 over x.

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We’re given that f of x minus 5 equals 1/3, so substituting x minus 5 into the first function,

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we get f of x minus 5… equals x minus 5 minus 10…all over x minus 5… equals 1/3.

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X minus 5 minus 10 simplifies to x minus 15. So x minus 15 over x minus 5 equals 1 over 3.

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To eliminate the fractions on both sides,

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we need to multiply both sides by (x - 5) and 3… to arrive at 3 times (x - 15) = x - 5.

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We distribute the 3 so that 3x - 45 = x – 5…and solve.

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And x equals 20. Our answer’s E.

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And our messy functions are no more. Thank you, Mr. Clean.

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