document.write( "Question 252190: Suppose f and g are linear functions such that f (g(t )) = t for all real t .
\n" ); document.write( "Find the rule for f (x) if f (5) = 3 and g(2) = 3 . \n" ); document.write( "

Algebra.Com's Answer #183983 by drk(1908) $\"\"$ $\"About$

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This may be a longer way than what you want . . . \r
\n" ); document.write( "\n" ); document.write( "Let
\n" ); document.write( "(i) F(t) = at + b
\n" ); document.write( "(ii) G(t) = ct + d\r
\n" ); document.write( "\n" ); document.write( "F(G(t)) = act + ad + b = t
\n" ); document.write( "We know two things from this: (1) ad = -b ; (2) a = 1/c.\r
\n" ); document.write( "\n" ); document.write( "By substitution, into (i) and (ii), we get
\n" ); document.write( "(iii) F(t) = (1/c)t - (1/c)d
\n" ); document.write( "(iv) G(t) = ct + d\r
\n" ); document.write( "\n" ); document.write( "substituting our coordinates, we get\r
\n" ); document.write( "\n" ); document.write( "(v) 3 = (1/c)*5 - d/c --> 3c = 5 - d
\n" ); document.write( "(vi) 3 = 2c + d --> 3 = 2c + d\r
\n" ); document.write( "\n" ); document.write( "adding these together, we get\r
\n" ); document.write( "\n" ); document.write( "3c + 3 = 5 + 2c. Solving for c, we get c = 2. Then from (vi), d = -1.\r
\n" ); document.write( "\n" ); document.write( "Now from (iii), we have F(t) = (1/2)t + (1/2). \n" ); document.write( "

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