Consider the composite function [tex]g(f(x))=x \sqrt{6}[/tex]. If [tex]f(x)=3 x^2[/tex], what is [tex]g(x)[/tex]?
A. [tex]g(x)=\sqrt{2 x}[/tex]
B. [tex]g(x)=\sqrt{x+3}[/tex]
C. [tex]g(x)=\sqrt{6 x}[/tex]
D. [tex]g(x)=\sqrt{9-x}[/tex]
Answer (1)
Let y = f ( x ) = 3 x 2 , then express x in terms of y : x = s q r t 3 y .
Substitute x into g ( f ( x )) = x s q r t 6 to get g ( y ) = s q r t 3 y s q r t 6 .
Simplify the expression: g ( y ) = s q r t 2 y .
Replace y with x to find g ( x ) = s q r t 2 x . The final answer is g ( x ) = 2 x .
Explanation
Understanding the Problem We are given the composite function g ( f ( x )) = x s q r t 6 and f ( x ) = 3 x 2 . Our goal is to find the expression for g ( x ) .
Expressing x in terms of y Let y = f ( x ) . Then y = 3 x 2 . We want to express x in terms of y . Dividing both sides by 3, we get x 2 = f r a c y 3 . Taking the square root of both sides (assuming 0"> x > 0 ), we have x = s q r t 3 y .
Finding g(y) Now we substitute f ( x ) = y into the composite function g ( f ( x )) = x s q r t 6 to get g ( y ) = x s q r t 6 . Since x = s q r t 3 y , we can substitute this expression for x into the equation for g ( y ) : g ( y ) = s q r t 3 y s q r t 6 = s q r t 3 6 y = s q r t 2 y .
Finding g(x) Finally, we replace y with x to find the expression for g ( x ) : g ( x ) = s q r t 2 x .
Conclusion Therefore, the function g ( x ) is 2 x .
Examples
Composite functions are useful in many real-world scenarios. For example, consider a store that marks up the price of an item by 50%, and then applies a 10% discount for a sale. If f ( x ) = 1.5 x represents the markup and g ( x ) = 0.9 x represents the discount, then the final price is g ( f ( x )) = 0.9 ( 1.5 x ) = 1.35 x . This means the final price is 35% higher than the original price. Understanding composite functions helps in analyzing such multi-step processes.
