Give three examples of a function f R R with the following

Give three examples of a function f : R --> R with the following property:

for all x in R, f(x + 1) = f(x + 3)

For each example of function, justify why it has the required property.

Hint: the sine function has the property: sin(x + 0) = sin(x + 2pi) for all x in R.

1. The constant function,f(x)=C

f(x+1)=f(x+3)=C

Now we look for two functions of period 2

2. sin(ax) has period 2PI/a we need this to be 2 so, 2PI/a=2

So, a=PI

HEnce, f(x)=sin(PI x)

f(x+1)=sin(PI(x+1))=sin(PIx+PI)=-sin(PI x)

f(x+3)=sin(PI(x+3))=sin(PI x+3PI)=-sin(PIx)

3. Another is f(x)=cos(PIx)