Question

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 1

The correct Option is A

Explanation:

A function f(x) is said to be odd function if :

f(-x) = -f(x)

Now, If we cut the graph of an odd function along the y-axis and then reflect its even half in the x-axis first followed by the reflection in the y-axis then The graph of the function could be seen having rotational symmetry about the origin.

So, we can say graph of odd function f(x) has rotational symmetry about the origin.

Hence, The correct Option is A

Answer 2

The correct answer is A) It has rotational symmetry about the origin.

First of all, let's explain the concept of odd functions. In principle, the graph of an odd function is symmetric with respect to the origin. To test this:


`If \ f(x) \ is \ odd \ function`

For each
`x` in the domain of
`f` then:


`f(x)=-f(-x)`

Let's take an example:

The following function:


`g(x)=x^(3)-x` is odd because:


`g(-x)=-g(x)` as follows:


`g(-x)=(-x)^(3)-(-x) \\ \therefore g(-x)=-x^(3)+x \\ \therefore g(-x)=-(x^(3)-x)=-g(x)`

Now a function has Rotational Symmetry when it still looks the same after a rotation. So let's take two functions, namely:


`f_(1)(x)=x^3 \\ \\ f_(2)(x)=sin(x)`

The graph has been rotated about the origin 180° for each function, so you can see that by doing this the graph is the same. In conclusion, an odd function is symmetric with respect to the origin and has rotational symmetry about the origin.