\( \DeclareMathOperator{cosec}{cosec} \)

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Here are some exam-style questions on this statement:

- "
*A function \( f \) is defined by \( f(x) = \arccos\left(\frac{x^2}{1-x^4}\right) \), \( x \in \mathbb{R} \), \( 0.785 \lt |x| \lt 5\).*" ... more - "
*Consider the function \(f(x) = \dfrac{x^3+2x}{5}, \; x \in \mathbb{R}\).*" ... more

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An even function is symmetric about the y-axis of a graph, while an odd function has rotational symmetry of order 2 about the origin.

The key formulae for even and odd functions are:

$$\begin{aligned}&\text{Even function: } f(x) = f(-x)\\&\text{Odd function: } f(x) = -f(-x)\end{aligned}$$

Here's an example of an odd function:

$$f(x) = x^3 + 2x$$

To prove that it's an odd function, we need to show that $$f(-x) = -f(x)$$. Therefore, $$\begin{aligned}f(-x)&= (-x)^3 + 2(-x)\\&= -x^3 - 2x\\&= -(x^3 + 2x)\\&= -f(x)\end{aligned}$$

Therefore, $$f(x)$$ is an odd function.

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