Answer:
Here we have:
g(x) = 3*f( 2*x - 2) - 1
We want to know the horizontal translation used here.
We need to discuss two types of transformations here.
horizontal translation.
If we have a function f(x), a horizontal translation of N units is written as:
g(x) = f(x + N)
if N is negative, the translation is to the right.
if N is positive, the translation is to the left.
Horizontal dilation.
For a function f(x) an horizontal dilation of scale factor k is written as:
g(x) = f( x/k)
You can see that in our function we have both of these transformations.
Because we compare the translation with f(x), we assume that the horizontal translation is the first thing we did, then we must have
translation:
g(x) = f(x + N)
and now a horizontal dilation of scale factor k
g(x) = f( (x + N)/k)
Then we must have that the argument of the above function is the same as the argument of the given function.
(x + N)/k = 2*x - 2
then:
k = 1/2
(x + N)/1/2 = 2*x + 2*N = 2*x - 2
2*N = -2
N = -2/2 = -1
This means that we have a translation of 1 unit to the right.
