Consider the composite function [tex]f(f(x))=\sqrt{\frac{1}{x^2-3}}[/tex]. If [tex]g(x)=x^2-3[/tex], what is [tex]f(x)[/tex]?
[tex]f(x)=\sqrt{\frac{-1}{x}}[/tex]
[tex]f(x)=\sqrt{\frac{-1}{x+6}}[/tex]
[tex]f(x)=\sqrt{\frac{1}{x}}[/tex]
[tex]f(x)=\sqrt{\frac{1}{x+6}}[/tex]
Answer (1)
We analyze the given composite function f ( f ( x )) = x 2 − 3 1 .
We test each of the provided options for f ( x ) by computing f ( f ( x )) .
We compare the result of each option with the given composite function.
None of the options match the given composite function, suggesting a possible error in the question or options.
Explanation
Understanding the Problem We are given the composite function f ( f ( x )) = x 2 − 3 1 and g ( x ) = x 2 − 3 . We need to find the expression for f ( x ) from the given options.
Listing the Options The given options for f ( x ) are:
f ( x ) = x − 1
f ( x ) = x + 6 − 1
f ( x ) = x 1
f ( x ) = x + 6 1
Solution Strategy We will test each option by computing f ( f ( x )) and comparing it to the given expression x 2 − 3 1 .
Testing Option 1 Option 1: f ( x ) = x − 1 . Then f ( f ( x )) = f ( x − 1 ) = x − 1 − 1 = − − 1 x = − i x This does not match the given f ( f ( x )) .
Testing Option 2 Option 2: f ( x ) = x + 6 − 1 . Then f ( f ( x )) = f ( x + 6 − 1 ) = x + 6 − 1 + 6 − 1 = − − x − 6 This does not match the given f ( f ( x )) .
Testing Option 3 Option 3: f ( x ) = x 1 . Then f ( f ( x )) = f ( x 1 ) = x 1 1 = 4 x This does not match the given f ( f ( x )) .
Testing Option 4 Option 4: f ( x ) = x + 6 1 . Then f ( f ( x )) = f ( x + 6 1 ) = x + 6 1 + 6 1 This does not match the given f ( f ( x )) .
Conclusion After testing all the options, none of them match the given composite function f ( f ( x )) = x 2 − 3 1 . There might be a typo in the question or the options.
Examples
Composite functions are used in various fields like computer graphics for transformations, in calculus for the chain rule, and in real-world scenarios like nested functions in programming. For example, consider a discount function applied after a tax function on a product's price. Understanding composite functions helps in analyzing such multi-layered processes.
