document.write( "Question 1194815: Let 𝑓 = {(3, 8), (2, 5), (4, −5), (9, 3 )} and 𝑔 = {(9, 2), (−5, 3), (5, 9 ), (8; 10), (1; 9)}. Determine the
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document.write( "compositions (𝑓 ∘ 𝑔) and (𝑔 ∘ 𝑓) if they exist.
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document.write( "1) (𝑓 ∘ 𝑔)
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document.write( "2) (𝑔 ∘ 𝑓) \n" );
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Algebra.Com's Answer #827084 by math_tutor2020(3817)![]() ![]() ![]() You can put this solution on YOUR website! \n" ); document.write( "I'll do question number 1 to get you started.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "The notation (𝑓 ∘ 𝑔)(x) is the same as f(g(x))\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "We have g(x) as the inner function, which means it goes first in the computation order.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "The first x coordinate mentioned in g(x) is x = 9 \n" ); document.write( "It's part of the ordered pair (9,2)\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Since the input x = 9 leads to the output y = g(x) = 2, this means \n" ); document.write( "f(g(9)) = f(2)\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Then we look through f(x) to see if there's an ordered pair with x = 2 \n" ); document.write( "There is such a point and it is (2,5)\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "f(g(9)) = f(2) = 5 \n" ); document.write( "f(g(9)) = 5 \n" ); document.write( "(𝑓 ∘ 𝑔)(9) = 5\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Therefore, (9,5) is one ordered pair in the function (𝑓 ∘ 𝑔)(x)\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "The next point mentioned in g(x) is (-5,3) \n" ); document.write( "x = -5 leads to y = g(x) = 3 \n" ); document.write( "f(g(-5)) = f(3) = 8 \n" ); document.write( "(𝑓 ∘ 𝑔)(-5) = 8\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "So (-5,8) is another point in the final answer.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Repeat this process for the remaining other x coordinates of the points in g(x). I'll let you do these steps.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Take note that \n" ); document.write( "x = 8 leads to g(x) = 10 \n" ); document.write( "f(g(8)) = f(10) = undefined \n" ); document.write( "because there isn't a point in f(x) with an x coordinate of 10. \n" ); document.write( "Therefore, x = 8 won't be mentioned in the final answer. We would say that x = 8 is not in the domain of 𝑓 ∘ 𝑔\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "------------------------------------\r \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Final answer for question 1: \n" ); document.write( "{ (9,5), (-5, 8), (5, 3), (1, 3) } \n" ); document.write( " \n" ); document.write( " \n" ); document.write( " |