document.write( "Question 403290: Suppose you have a function y = f(x) such that the domain of f(x) is 1 ≤ x ≤ 6 and the range of f(x) is −3 ≤ y ≤ 5. What is the domain of f(2(x − 3)) \n" ); document.write( "

Algebra.Com's Answer #285363 by robertb(5830) $\"\"$ $\"About$

You can put this solution on YOUR website!
The domain of f(2(x-3))can be determined in all generality, not only using discrete values.
\n" ); document.write( "Let g(x) = 2(x-3) = 2x - 6.
\n" ); document.write( "Then f(2(x-3)) = (f o g)(x) = f(g(x)). Let the domain of g be $\"D%5Bg%5D\"$ , its range $\"R%5Bg%5D\"$ . Similarly let the domain of f be $\"D%5Bf%5D\"$ , its range $\"R%5Bf%5D\"$ .
\n" ); document.write( "Then to find the domain of f o g we must find all x values in $\"D%5Bg%5D\"$ (which is the set of all real numbers), that are the pullback of $\"R%5Bg%5D\"$ intersection $\"D%5Bf%5D\"$ . Since the range of g is represented by 2(x-3), and the domain of f is the closed interval [1,6], we must then have
\n" ); document.write( " $\"1+%3C=+2%28x+-+3%29+%3C=+6\"$
\n" ); document.write( "<==> $\"1%2F2+%3C=+x+%0D%0A-+3+%3C=+3\"$
\n" ); document.write( "<==> $\"7%2F2+%3C=+x+%3C=+6\"$ .
\n" ); document.write( "Thus the domain of (f o g)(x) = f(2(x-3)) is the closed interval [7/2, 6]. \n" ); document.write( "

\n" );