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International Baccalaureate Mathematics

Functions

Syllabus Content

Odd and even functions.
Finding the inverse function, f-1(x), including domain restriction.
Self-inverse functions.

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Furthermore

Official Guidance, clarification and syllabus links:

Even: \(f(-x)=f(x)\)

Odd: \(f(-x)=-f(x)\)

Includes periodic functions.

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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