Consecutive vertical and horizontal transformations of a function

[pluto1]

Dear all,
I am stuck on this question:
"If f(x)=(2^x)+1, give in simplest terms the formula for h(x), which is obtained from transforming f(x) by

a vertical stretch, scale factor 8 relative to y=0
a translation by vector (1,4)
a horizontal stretch, scale factor 1/2 relative to x=0"

This is what I understand from the question:

A vertical stretch by scale factor 8 means h(x)= 8 f(x)
A translation by vector (1,4) means h(x)= f(x-1) + 4
A horizontal stretch by scale factor 1/2 means h(x)= f(1/2x)
Where the horizontal shift is applied prior to the horizontal stretch.

This gives me

h(x)= 8 ((2^(1/2 x-1) + 1/2 x - 1))+ 4
h(x)= 2^(2x-2) + 16 x - 4

h(x)=4^(x+1)+16x-4 according to the solutions at the back of my textbook. I would really appreciate some help. Thank you so much in advance.

 

[HallsofIvy]

In what order are you doing these transformations?

 

[pluto1]

I applied the horizontal changes before the vertical changes, with the translation by -1 prior to the stretch by 1/2

I just tried the opposite with all the vertical changes first but that does not give me the correct solution either