The function g(x) is written in a confusing way.
The most logical form for g(x) according to the parent fucntion and the statements is this:
g(x) = [(-1/2)x]³
So, I will answer the question with such g(x).
And I will explain each step such that this answer is useful for you.
Statetements:
a) The graph passes through the origin → True
The origin is the point (0,0)
Then plug in 0 into g(x).
The result is [ (-1/2) (0)] ³, which is 0.
So, indeed the point (0,0), the origin is in the graph.
b) As x approaches negative infinity, the graph of g(x) approaches infinity → TRUE
As x approaches negative infinity the denominator g(x) becomes greater and greater.
Try this: g(-10) g(-100), g(-1000).
g(-10) = 5³
g(-100) = (50)³
g(-1000) = (500)³
That shows you the trend: g(x) approaches infinity when x approaches to negative infinity.
c) As x approaches infinity, the graph of g(x) approaches infinity.→ False
As x approaches infinity, the graph of g(x) approaches negative infinity.
For example g(20000) = [- (20000/2) ]³ = - (10000)³
And as x grows g(x) becomes more negative.
d) The domain of the function is all real numbers → TRUE
The function g(x) accepst any value of x.
e) The range of the function is all real numbers → TRUE
g(x) goes from - infinity to + infinity and is continuous.
f) The graph of the function has three distinct zeros → FALSE
The only zero of g(x) is for x = 0.
