International Baccalaureate Mathematics

Syllabus Content

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

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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Functions A relationship between two sets can be called a mapping. Elements of the first set (domain) are mapped to elements of the second set (range). A function is a special type of mapping for which one value in the domain maps to one, and only one value in the range.Pupils in Primary school will use the concept of function machines to perform calculations. They will then learn to ‘work backwards’ to find the inverse function. The study of functions becomes more formal as pupils become more proficient and able to cope with more complex mathematical ideas.

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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