Which of the following is an odd function?
[tex]f(x)=x^3+5 x^2+x[/tex]
[tex]f(x)=\sqrt{x}[/tex]
[tex]f(x)=x^2+x[/tex]
[tex]f(x)=-x[/tex]
Answer (2)
An odd function satisfies f ( − x ) = − f ( x ) .
Test f ( x ) = x 3 + 5 x 2 + x : f ( − x ) = − x 3 + 5 x 2 − x = − f ( x ) .
Test f ( x ) = x : Not defined for x < 0 , so not odd.
Test f ( x ) = x 2 + x : f ( − x ) = x 2 − x = − f ( x ) .
Test f ( x ) = − x : f ( − x ) = x = − f ( x ) .
Therefore, the odd function is f ( x ) = − x .
Explanation
Understanding Odd Functions An odd function is a function that satisfies the condition f ( − x ) = − f ( x ) for all x in the domain of f . We are given four functions: f ( x ) = x 3 + 5 x 2 + x , f ( x ) = x , f ( x ) = x 2 + x , and f ( x ) = − x . We need to determine which of these functions is odd.
Checking the First Function For f ( x ) = x 3 + 5 x 2 + x , we need to check if f ( − x ) = − f ( x ) . Let's compute f ( − x ) : f ( − x ) = ( − x ) 3 + 5 ( − x ) 2 + ( − x ) = − x 3 + 5 x 2 − x Now, let's compute − f ( x ) : − f ( x ) = − ( x 3 + 5 x 2 + x ) = − x 3 − 5 x 2 − x Since f ( − x ) = − f ( x ) , the function f ( x ) = x 3 + 5 x 2 + x is not odd.
Checking the Second Function For f ( x ) = x , the domain of the function is x ≥ 0 . If we consider f ( − x ) = − x , this is only defined for x ≤ 0 . For 0"> x > 0 , f ( − x ) is not a real number. Therefore, f ( x ) = x is not an odd function.
Checking the Third Function For f ( x ) = x 2 + x , we need to check if f ( − x ) = − f ( x ) . Let's compute f ( − x ) : f ( − x ) = ( − x ) 2 + ( − x ) = x 2 − x Now, let's compute − f ( x ) : − f ( x ) = − ( x 2 + x ) = − x 2 − x Since f ( − x ) = − f ( x ) , the function f ( x ) = x 2 + x is not odd.
Checking the Fourth Function For f ( x ) = − x , we need to check if f ( − x ) = − f ( x ) . Let's compute f ( − x ) : f ( − x ) = − ( − x ) = x Now, let's compute − f ( x ) : − f ( x ) = − ( − x ) = x Since f ( − x ) = − f ( x ) , the function f ( x ) = − x is an odd function.
Final Answer Therefore, the odd function among the given options is f ( x ) = − x .
Examples
Odd functions are symmetric about the origin. In physics, they can describe certain types of waves or signals where the negative part mirrors the positive part. For example, the sine function, sin(x), is an odd function and is used to model alternating current. Understanding odd functions helps in simplifying calculations and analyzing symmetrical phenomena in various fields.
The odd function among the given options is f ( x ) = − x , as it satisfies the condition f ( − x ) = − f ( x ) . The other functions do not fulfill this criterion. Thus, the correct choice is f ( x ) = − x .
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