## Sketching Periodic functions with GC Suppose we want to sketch the graph from -3 to 8.

1. Enter by pressing “Math, Piecewise”. 2. Adjust the window setting to the required domain. 3. Trace the end point at -3 since it is not 1 complete cycle. ## Finding functions that form a particular composite function  ## Solving function f = f inverse      Tip: If we are not sure whether f and f inverse intersect, we can use the graphing calculator TI 84 plus to plot function f and f inverse together.

Y1 = “function f”

DrawInv Y1 (2nd prgm 8, alpha trace, Y1)

## Challenging Functions problem 3 Alternative method ## Challenging functions problem 1

Functions f and g are defined as follows:  Solution    Therefore, both functions f and g repeat itself.

1994 divide by 3 has a remainder of 2. Therefore, 2011 divide by 3 has a remainder of 1. Therefore, ## Difference between f(f-inverse) and (f-inverse)f

Some questions on functions require students to understand the difference between  ff-1 and f-1f. Though both = x, they are two different functions because their domains might be different.

Therefore,

Example f(x) = x+2, x> 0  ## 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. 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. 