document.write( "Question 1124192: The function f(x)=3^x is an exponential function and is often referred to as the \"tripling function.\" \r
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document.write( "a. for any value of x, f(x+1)is how many times as large as f(x).\r
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document.write( "b. for any value of x, f(x+2) is how many times as large as f(x).\r
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document.write( "c. If x increases by 4, f(x) becomes how many times as large?
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Algebra.Com's Answer #740551 by Theo(13342)![]() ![]() You can put this solution on YOUR website! f(x) = 3^x\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "the value of f(x) is whown in the following table.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \r\n" ); document.write( "\r\n" ); document.write( "x f(x)\r\n" ); document.write( "0 1\r\n" ); document.write( "1 3\r\n" ); document.write( "2 9\r\n" ); document.write( "3 27\r\n" ); document.write( "4 82\r\n" ); document.write( "\r\n" ); document.write( "\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "as can be seen, each time x goes up by 1, the value of f(x) goes up by a factor of 3.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "take any random value of x and you'll see this tripling IN action.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "i chose x = 7 \n" ); document.write( "that's a large number and will lead to equally large numbers.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "f(x) = f(7) = 3^7 = 2187 \n" ); document.write( "f(x+1) = f(8) = 3^8 = 6561 = 3 * 2187 = 3^1 * 2187 \n" ); document.write( "f(x+2) = f(9) = 3^9 = 19683 3 * 6561 = 3 * 3 * 2187 = 3^2 * 2187 \n" ); document.write( "f(x+3) = f(10) = 3^10 = 59049 = 3 * 19683 = 3 * 3 * 3 * 2187 = 3^3 * 2187 \n" ); document.write( "f(x+4) = f(11) = 3^11 = 177147.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "so, f(x + n) can be expected to be 3^n * 2187, if f(x) = 7\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "in general, then:\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "f(x + n) can be expected to be 3^n * f(x), for any value of x.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "this works for any positive value of x.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "take any random value that can be calculated, and you should see that it's true.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "for example, i randomly chose x = 15.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "f(15) = 3^15 = 14348907 \n" ); document.write( "f(15+5) = 3^20) = 3486784401 \n" ); document.write( "f(15+5) = 3^15 * 3^5 = 3486784401 = 3^20\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "but what happens when x is negative?\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "a simple example might show what happens.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "let x = -3 \n" ); document.write( "then f(x) = x^-3 = 1/27\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "f(x+3) would be equal to f(-3+3) = f(0) which is equal to 1.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "f(x+3) is also equal to f(-3) * 3^3 = 1/27 * 27 = 1\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "1 is equal to 27 * 1/27, therefore the formula holds.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "another example:\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "f(-9) = 3^-9 = 5.080526343 * 10^-5\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "f(-9+5) = f(-9) * f(5) which is equal to 3^-9 * 3^5 which is equal to .012345679.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "f(-9+5) is also equal to f(-4) which is equal to .012345679.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( ".012345679 divided by 5.080526343 * 10^-5 is equal to 243.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "3^5 is equal to 243.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "therefore, the relationship holds for positive and negative values of x.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "consequently, f(x+n) is equal to f(x) * f(n).\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "in case it got lost in the discussion, however, the answer to your questions is:\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "a. for any value of x, f(x+1)is how many times as large as f(x). \r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "f(x+1) is 3^1 times as large as f(x) = 3 times as large.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "b. for any value of x, f(x+2) is how many times as large as f(x). \r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "f(x+2) is 3^2 times as large as f(x) = 9 times as large.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "c. If x increases by 4, f(x) becomes how many times as large?\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "f(x+4) is 3^4 times as large as f(x) = 81 times as large.\r \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " |