Signals and Systems-Time Scaling and Time Shifting

I have find this part as the most confusing one among the students.
Though everyone feels that it is easy (which indeed is true), yet mistakes are often commited
Take a look at the following note:

 Time Shifting and time scaling of x(t) to y(t) = x(at+b)

 It is better to first time shift and then time scale because time shift is w.r.t to “t” and not “at”

Take the following example:-

Suppose we are provided with x(t) and we have to find y(t) = x(2t+3)

A step by step procedure is shown below:-

Take another example of a GATE EE 2010 question:

 Q.  Given f(t) and g(t) as shown below:

The simplest approach is:-

Take g(t) = f((at + b),

From graph, g(3) = 1 = f(a.3 + b) and g(5) = 1 = f(a.5 = b)

We can see from graphs that left step of f(t) give left step of g(t) and right step of f(t) gives right step ofg(t).

Hence we have:

a.3 + b = 0  and a.5 + b = 1 which yield ,

a =1/2 and b=-3/2

Therefore g(t) = f(t/2 – 3/2). Ans (d).