Which of the following is an odd function?
[tex]g(x)=x^2[/tex]
[tex]g(x)=5 x-1[/tex]
[tex]g(x)=3[/tex]
[tex]g(x)=4 x[/tex]
Answer (2)
A function g ( x ) is odd if g ( − x ) = − g ( x ) .
Test g ( x ) = x 2 : g ( − x ) = ( − x ) 2 = x 2 , which is even.
Test g ( x ) = 5 x − 1 : g ( − x ) = − 5 x − 1 , which is neither even nor odd.
Test g ( x ) = 3 : g ( − x ) = 3 , which is even.
Test g ( x ) = 4 x : g ( − x ) = − 4 x , which is odd.
The odd function is 4 x .
Explanation
Understanding Odd Functions We are given four functions and need to identify the odd function among them. A function g ( x ) is odd if it satisfies the condition g ( − x ) = − g ( x ) for all x in its domain. Let's examine each function.
Analyzing g(x) = x^2
g ( x ) = x 2 :
We evaluate g ( − x ) = ( − x ) 2 = x 2 . Since g ( − x ) = g ( x ) , this function is even, not odd.
Analyzing g(x) = 5x - 1
g ( x ) = 5 x − 1 :
We evaluate g ( − x ) = 5 ( − x ) − 1 = − 5 x − 1 . Then, we find − g ( x ) = − ( 5 x − 1 ) = − 5 x + 1 . Since g ( − x ) e q − g ( x ) , this function is neither even nor odd.
Analyzing g(x) = 3
g ( x ) = 3 :
We evaluate g ( − x ) = 3 . Then, we find − g ( x ) = − 3 . Since g ( − x ) e q − g ( x ) , this function is even, not odd.
Analyzing g(x) = 4x
g ( x ) = 4 x :
We evaluate g ( − x ) = 4 ( − x ) = − 4 x . Then, we find − g ( x ) = − 4 x . Since g ( − x ) = − g ( x ) , this function is odd.
Conclusion Therefore, the odd function among the given options is g ( x ) = 4 x .
Examples
Odd functions are symmetric about the origin. In physics, they can describe certain types of waves or signals where the negative part of the signal mirrors the positive part. For example, if f ( x ) represents the displacement of a wave at position x , and f ( x ) is an odd function, then the displacement at − x is the negative of the displacement at x . This symmetry simplifies many calculations and helps in understanding the behavior of these systems.
The odd function among the given options is g ( x ) = 4 x . An odd function satisfies the condition g ( − x ) = − g ( x ) . The analysis shows that the function meets this condition, while the others do not.
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