If $f(x)$ is an odd function, which statement about the graph of $f(x)$ must be true?
A. It has rotational symmetry about the origin.
B. It has line symmetry about the line $y=-x$.
C. It has line symmetry about the $y$-axis.
D. It has line symmetry about the $x$-axis.
Answer (2)
Odd functions are defined by the property f ( − x ) = − f ( x ) , which implies that their graphs have rotational symmetry about the origin. They do not possess line symmetry about the y-axis, x-axis, or the line y = − x . Therefore, the correct choice is: A. It has rotational symmetry about the origin.
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An odd function satisfies f ( − x ) = − f ( x ) .
This property implies rotational symmetry about the origin.
Line symmetry about the y-axis corresponds to even functions, where f ( x ) = f ( − x ) .
Line symmetry about the x-axis implies f ( x ) = 0 .
Therefore, the correct statement is that an odd function has rotational symmetry about the origin. It has rotational symmetry about the origin.
Explanation
Understanding Odd Functions An odd function is defined by the property that f ( − x ) = − f ( x ) for all x in the domain of f . This property implies that the graph of an odd function has rotational symmetry about the origin. Let's analyze why this is the case and why the other options are not necessarily true.
Rotational Symmetry Rotational symmetry about the origin means that if you rotate the graph of the function by 180 degrees about the origin, you get the same graph back. This is exactly what the property f ( − x ) = − f ( x ) describes. For any point ( x , f ( x )) on the graph, the point ( − x , − f ( x )) is also on the graph. This corresponds to a 180-degree rotation about the origin.
Line Symmetry about the y-axis Line symmetry about the y -axis means that the function is even, i.e., f ( x ) = f ( − x ) . This is not true for odd functions (except for the trivial case f ( x ) = 0 ).
Line Symmetry about the x-axis Line symmetry about the x -axis means that if ( x , y ) is on the graph, then ( x , − y ) is also on the graph. This would imply f ( x ) = − f ( x ) , which is only true if f ( x ) = 0 .
Line Symmetry about y = -x Line symmetry about the line y = − x means that if ( x , y ) is on the graph, then ( − y , − x ) is also on the graph. This is not a general property of odd functions. For example, f ( x ) = x is an odd function, and it has line symmetry about y = x and y = − x . However, f ( x ) = x 3 is an odd function, but it does not have line symmetry about the line y = − x .
Conclusion Therefore, the only statement that must be true for all odd functions is that it has rotational symmetry about the origin.
Examples
Odd functions are used in physics to describe certain types of symmetry. For example, in signal processing, the Fourier transform of a real-valued odd function is purely imaginary. In mechanics, potential energy functions are often even, while forces, which are derivatives of potential energy, are often odd. Understanding the properties of odd functions helps in simplifying calculations and understanding the underlying symmetries in various physical systems.
