Consider a real function $g(t)$. Now consider another real function $f(t) = -t +3$ that transforms the domain of $g(t)$. Suppose I have the graph of $g(t)$ and I'm trying to plot the $g(f(t))$ graph by drawing. In order to get the correct graph, I could think of $f(t) = -(t-3)$, in which case, to plot $g(f(t))$, I should first invert $g(t)$ with respect to the vertical axis and then shift $3$ units to the right.The problem is that I'm not seeing the intuition into inverting $t$ first, and then shifting the inverted graph by 3 units to the right. Why is thinking of shifting 3 units to the right first $(t-3)$ and then inverting the shifted graph $-(t-3)$, a wrong procedure? Why is thinking of inverting the domain $( -t )$ and then shifting the inverted graph 3 units to the left $( -t + 3 )$ a wrong procedure ?
I got the intuitive feeling for all transformations, both of domain and of the range.This one though, is where I fall into confusion. Thanks