Answer:
The correct option is the fourth one.
Step-by-step explanation:
FIRST QUESTION:
To solve this problems we need to know the following things:
1. Given f(x) = k*g(x). We know that the graph of f(x) will be identical to the graph of g(x) but enlarged or compressed depending on the value of k.
2. Given f(x) = g(x) + k. We know that the graph of f(x) will be identical to the graph of g(x) but shift downwards if k<0 or shift upwards if k>0.
Having said this, we have that:
f(x) = 1/2 * log_2 (x) - 8
In this case, there are two transformations: The function is supressed by 1/2 and also shifted downwards by 8 units.
SECOND QUESTION
We need to find the portion of domain where f(x) = x^2 + a is one-to-one.
A function f is one-to-one if each x in the domain has exactly one image in the range. And, no y in the range is the image of more than one x in the domain.
So if we want a one-to-one function we need to restrict the domain starting from x=0. So it would be: [0, inf)
Now to find the inverse function, we need to solve the equation for "x"
y = x^2 + a
y - a = x^2
x = sqrt(y-a)
Then, the inverse function would be:
y = sqrt(x-a)
The correct option is the fourth one.
