If f(x) is an odd function, which statement about the graph of f(x) must be true?

Asked by maham237 @ 18/12/2020 in Mathematics viewed by 405 People

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.

Answered by maham237 @ 18/12/2020

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.

If f(x) is an odd function, which statement about the graph of f(x) must be true?

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