Danika concludes that the following functions are inverses of each other because [tex]f(g(x))=x[/tex]. Do you agree with Danika? Explain your reasoning.
[tex]f(x)=|x|[/tex]
[tex]g(x)=-x[/tex]
Answer (2)
No, I do not agree with Danika. N o
Explanation
Analyzing the Problem Let's analyze the given functions and Danika's conclusion. We have f ( x ) = ∣ x ∣ and g ( x ) = − x . Danika claims that these functions are inverses because f ( g ( x )) = x . We need to verify if this is correct.
Conditions for Inverse Functions To check if two functions are inverses, we need to verify two conditions:
f ( g ( x )) = x for all x in the domain of g .
g ( f ( x )) = x for all x in the domain of f .
If both conditions are met, then the functions are inverses. If at least one condition is not met, they are not inverses.
Computing f(g(x)) Let's compute f ( g ( x )) . We substitute g ( x ) = − x into f ( x ) = ∣ x ∣ :
f ( g ( x )) = f ( − x ) = ∣ − x ∣ = ∣ x ∣ .
So, f ( g ( x )) = ∣ x ∣ .
Computing g(f(x)) Now let's compute g ( f ( x )) . We substitute f ( x ) = ∣ x ∣ into g ( x ) = − x :
g ( f ( x )) = g ( ∣ x ∣ ) = − ∣ x ∣ .
So, g ( f ( x )) = − ∣ x ∣ .
Analyzing the Results Now let's analyze the results. We have f ( g ( x )) = ∣ x ∣ . For f ( g ( x )) to be equal to x , we need ∣ x ∣ = x . This is only true when x ≥ 0 . If x < 0 , then ∣ x ∣ = − x , so f ( g ( x )) = x for all x . Also, we have g ( f ( x )) = − ∣ x ∣ . For g ( f ( x )) to be equal to x , we need − ∣ x ∣ = x . This is only true when x ≤ 0 . If 0"> x > 0 , then − ∣ x ∣ = − x , so g ( f ( x )) = x for all x .
Conclusion Since f ( g ( x )) = ∣ x ∣ and g ( f ( x )) = − ∣ x ∣ , the conditions for inverse functions are not met for all x . Therefore, f ( x ) = ∣ x ∣ and g ( x ) = − x are not inverses of each other. Danika's conclusion is incorrect because f ( g ( x )) = x is not true for all x in the domain.
Examples
In cryptography, inverse functions are crucial for encoding and decoding messages. If f ( x ) encodes a message, then its inverse f − 1 ( x ) decodes it. If f ( x ) and g ( x ) were truly inverses, applying f and then g would return the original message. However, since ∣ − x ∣ doesn't always give us back x , these functions can't reliably encode and decode information in a reversible way.
The functions f ( x ) = ∣ x ∣ and g ( x ) = − x are not inverses of each other since f ( g ( x )) = ∣ x ∣ and g ( f ( x )) = − ∣ x ∣ , which do not equal x for all values of x . Danika's conclusion is incorrect.
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