Periodic, even and odd functions

H2 math syllabus 9740 did not state that periodic functions are included in the syllabus. However, questions on periodic functions have appeared in A level 2009 and 2013.

So it is good to learn periodic, even and odd functions.

Periodic functions

A periodic function has a graph with a basic pattern that repeats at regular intervals.

For example, sin x is periodic and its period is 2 pi.

sin(x) = sin(x+ 2pi)

For  a periodic function with period a

f(x) = f(x+a) = f(x+2a) = ..

Simiarly

f(x) = f(x-a) = f(x-2a) = …

i.e. f(x) = f(x+ka) where k is an integer

Even functions

A function is said to be even if f(x) = f(-x) for all values of x. The graphs of all even functions are symmetrical about the vertical axis.

cos

 

Odd functions

A function is odd if f(x) = -f(-x) for all values of x. The graphs of all odd functions are symmetrical about the origin. One section of the graph can be rotated about the origin through 180 degrees to give the other section.

odd

 

Understanding Composite Functions

For the composite function fg,

Understanding composite functionsTo find the range of the composite function, there are 3 approaches

Method 1ex

Method 2 (Only for strictly increasing or decreasing function)

Since both function g and f are strictly increasing, we can deduce the range in the following manner:

composite

Method 3

Determine the range based on the rule and domain of the composite function

Using the same f and g definition above,

Tex2Img_1399597518

Tex2Img_1399597421

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