Check all of the functions that are odd.
f(x)=x^3-x^2
f(x)=x^5-3 x^3+2 x
f(x)=4 x+9
f(x)=\frac{1}{x}
Answer (2)
The odd functions are f ( x ) = x 5 − 3 x 3 + 2 x and f ( x ) = x 1 . The functions f ( x ) = x 3 − x 2 and f ( x ) = 4 x + 9 are not odd. To be odd, a function must satisfy f ( − x ) = − f ( x ) for all x .
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Check if f ( − x ) = − f ( x ) for each function.
For f ( x ) = x 3 − x 2 , f ( − x ) = − x 3 − x 2 and − f ( x ) = − x 3 + x 2 , so it's not odd.
For f ( x ) = x 5 − 3 x 3 + 2 x , f ( − x ) = − x 5 + 3 x 3 − 2 x and − f ( x ) = − x 5 + 3 x 3 − 2 x , so it's odd.
For f ( x ) = 4 x + 9 , f ( − x ) = − 4 x + 9 and − f ( x ) = − 4 x − 9 , so it's not odd.
For f ( x ) = x 1 , f ( − x ) = − x 1 and − f ( x ) = − x 1 , so it's odd.
The odd functions are: f ( x ) = x 5 − 3 x 3 + 2 x e wl in e f ( x ) = x 1
Explanation
Understanding Odd Functions We are given four functions and we need to determine which of them are odd. A function f ( x ) is odd if f ( − x ) = − f ( x ) for all x in the domain of f . Let's analyze each function.
Analyzing f(x) = x^3 - x^2
f ( x ) = x 3 − x 2 We compute f ( − x ) = ( − x ) 3 − ( − x ) 2 = − x 3 − x 2 . We also compute − f ( x ) = − ( x 3 − x 2 ) = − x 3 + x 2 .
Since f ( − x ) = − x 3 − x 2 and − f ( x ) = − x 3 + x 2 , we have f ( − x ) e q − f ( x ) . Therefore, f ( x ) = x 3 − x 2 is not an odd function.
Analyzing f(x) = x^5 - 3x^3 + 2x
f ( x ) = x 5 − 3 x 3 + 2 x We compute f ( − x ) = ( − x ) 5 − 3 ( − x ) 3 + 2 ( − x ) = − x 5 + 3 x 3 − 2 x .
We also compute − f ( x ) = − ( x 5 − 3 x 3 + 2 x ) = − x 5 + 3 x 3 − 2 x .
Since f ( − x ) = − x 5 + 3 x 3 − 2 x and − f ( x ) = − x 5 + 3 x 3 − 2 x , we have f ( − x ) = − f ( x ) . Therefore, f ( x ) = x 5 − 3 x 3 + 2 x is an odd function.
Analyzing f(x) = 4x + 9
f ( x ) = 4 x + 9 We compute f ( − x ) = 4 ( − x ) + 9 = − 4 x + 9 .
We also compute − f ( x ) = − ( 4 x + 9 ) = − 4 x − 9 .
Since f ( − x ) = − 4 x + 9 and − f ( x ) = − 4 x − 9 , we have f ( − x ) e q − f ( x ) . Therefore, f ( x ) = 4 x + 9 is not an odd function.
Analyzing f(x) = 1/x
f ( x ) = x 1 We compute f ( − x ) = ( − x ) 1 = − x 1 .
We also compute − f ( x ) = − x 1 .
Since f ( − x ) = − x 1 and − f ( x ) = − x 1 , we have f ( − x ) = − f ( x ) . Therefore, f ( x ) = x 1 is an odd function.
Final Answer In conclusion, the odd functions are f ( x ) = x 5 − 3 x 3 + 2 x and f ( x ) = x 1 .
Examples
Odd functions are symmetric about the origin, meaning if you rotate the graph 180 degrees about the origin, you get the same graph back. This property is useful in physics, for example, when analyzing signals or waves that have symmetry. In signal processing, if a signal is odd, it simplifies certain calculations, such as Fourier series expansions, making it easier to analyze the signal's frequency components. Also, in optics, the intensity profile of certain laser beams can be described by odd functions.
