Answer: A) reflection over the x-axis, plus a vertical translation
Step-by-step explanation:
Ok, when we have a function y = f(x)
> A reflection over the x-axis changes a point (x, y) to a point (x, -y), then for a function (x , y = f(x)) the point will change to (x, -y =- f(x))
then for a funtion g(x), this tranformation can be written as h(x) = -g(x).
> A vertical translation of A units (A positive) up for a function g(x) can be written as: h(x) = g(x) + A.
Then in this case we have:
y = g(x) = ln(x)
and the transformed function is h(x) = -ln(x) + 64
Then we can start with h(x) = g(x)
first do a reflection over the x-axis, and now we have:
h(x) = -g(x) = -ln(x)
And now we can do a vertical translation of 64 units up
h(x) = -g(x) + 64 = -ln(x) + 64
Then the correct option is:
A) reflection over the x-axis, plus a vertical translation
, so our solutions are 
and 
.