Mathematics

Problem

Let

h(x)=-2\,5^{x-3}+4

. Answer the following, showing each logical step clearly: a. (i) Rewrite

h(x)

as a sequence of vertical and horizontal transformations of the basic curve

y=5^{x}

. b. (ii) Determine the domain of

h(x)

. c. (iii) Obtain the exact range of

h(x)

. Give full reasoning, starting from the core property

5^{x}>0\;\forall x\in\mathbb{R}

. d. (iv) State the equation of the horizontal asymptote of the graph of

h(x)