(a) Assume g(x)=-(⅔4x+8)-5.
p(x)=ex could be regarded as the parent function, because it has the same basic shape.
If exln(b)≡bx, (16/81)x=exln(16/81)=e-1.622x. A negative coefficient on the exponent reflects the parent in the y (vertical) axis. In other words, e-1.622x is a reflection of e1.622x. So this is the early step in the transformation process. The magnitude of the coefficient produces a horizontal dilation. The transformation is ex→epx, where p=ln(16/81). The next part of the transformation is epx→aepx (dilation, or amplification, vertically), and the final part is a downward shift (relocation) caused by the negative value of c.
(b) g(x)=-(⅔)8(24/34)x-5=abx+c, so a=-(⅔)8=-256/6561, b=16/81, c=-5.
(c) The horizontal asymptote is y=-5, because the exponential term shrinks to zero as x increases, that is, b=16/81<1 so increasing powers make it smaller. e-1.622x→0 as x→∞.
(d) g(x) increases towards the asymptote.
(e) The domain is (-∞,∞), that is, it's unrestricted.
(f) The range is (-∞,-5). The asymptote is the upper limit to g(x).