Consider the composite function [tex]$f(g(x))=\sqrt{\frac{1}{x^2-3}}$[/tex]. If [tex]$g(x)=x^2-3$[/tex], what is [tex]$f(x)$[/tex]?
A. [tex]$f(x)=\sqrt{\frac{1}{x}}$[/tex]
B. [tex]$f(x)=\sqrt{\frac{1}{x+6}}$[/tex]
C. [tex]$f(x)=\sqrt{\frac{-1}{x}}$[/tex]
D. [tex]$f(x)=\sqrt{\frac{-1}{x+6}}$[/tex]
Answer (1)
Given f ( g ( x )) = x 2 − 3 1 and g ( x ) = x 2 − 3 .
Substitute x for g ( x ) in f ( g ( x )) .
Replace x 2 − 3 with x in the expression for f ( g ( x )) .
Therefore, f ( x ) = x 1 .
Explanation
Understanding the Problem We are given the composite function f ( g ( x )) = x 2 − 3 1 and g ( x ) = x 2 − 3 . Our goal is to find the function f ( x ) .
Finding f(x) Since f ( g ( x )) = x 2 − 3 1 and g ( x ) = x 2 − 3 , we can find f ( x ) by substituting x for g ( x ) in the expression for f ( g ( x )) . This means we replace every instance of x 2 − 3 in the expression for f ( g ( x )) with x .
Substituting x for g(x) So, f ( g ( x )) = f ( x 2 − 3 ) = x 2 − 3 1 . Replacing x 2 − 3 with x , we get f ( x ) = x 1 .
Final Answer Therefore, f ( x ) = x 1 .
Examples
Composite functions are useful in many real-world scenarios. For example, consider a store that offers a discount of 10% on all items and then applies a sales tax of 5%. If g ( x ) = 0.9 x represents the price after the discount and f ( x ) = 1.05 x represents the price after sales tax, then f ( g ( x )) represents the final price you pay. Understanding composite functions helps in analyzing such multi-step processes.
