The graph of the parent function f(x) = x3 is transformed such that g(x) = (–1/2x). Which statements about the graph of g(x) are accurate? Check all that apply.The graph passes through the origin.As x approaches negative infinity, the graph of g(x) approaches infinity.As x approaches infinity, the graph of g(x) approaches infinity.The domain of the function is all real numbers.The range of the function is all real numbers.The graph of the function has three distinct zeros.
Accepted Solution
A:
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