Question

the graph of the parent function $f(x) = x^3$ is transformed such that $g(x) = \left(-\frac{1}{2}x^3\
ight)$. which statements about the graph of $g(x)$ are accurate? select three options.\
\\(\square\\) the graph does not pass through the origin.\
\\(\square\\) as $x$ approaches negative infinity, the graph of $g(x)$ approaches infinity.\
\\(\square\\) as $x$ approaches infinity, the graph of $g(x)$ approaches infinity.\
\\(\square\\) the domain of the function is all real numbers.\
\\(\square\\) the range of the function is all real numbers.

Explanation:

Step1: Analyze \( g(x)=-\frac{1}{2}x^3 \) at \( x = 0 \)

Substitute \( x = 0 \) into \( g(x) \): \( g(0)=-\frac{1}{2}(0)^3 = 0 \). So the graph passes through the origin. Thus, "The graph does not pass through the origin" is false.

Step2: Analyze end - behavior as \( x\to-\infty \)

For the function \( y = x^3 \), as \( x\to-\infty \), \( y\to-\infty \). For \( g(x)=-\frac{1}{2}x^3 \), when \( x\to-\infty \), \( x^3\to-\infty \), and multiplying by \( -\frac{1}{2} \) (a negative number) gives \( g(x)\to\infty \). So "As \( x \) approaches negative infinity, the graph of \( g(x) \) approaches infinity" is true.

Step3: Analyze end - behavior as \( x\to\infty \)

For \( y = x^3 \), as \( x\to\infty \), \( y\to\infty \). For \( g(x)=-\frac{1}{2}x^3 \), when \( x\to\infty \), \( x^3\to\infty \), and multiplying by \( -\frac{1}{2} \) gives \( g(x)\to-\infty \). So "As \( x \) approaches infinity, the graph of \( g(x) \) approaches infinity" is false.

Step4: Analyze the domain of \( g(x) \)

The function \( g(x)=-\frac{1}{2}x^3 \) is a polynomial function. Polynomial functions are defined for all real numbers. So the domain of \( g(x) \) is all real numbers. Thus, "The domain of the function is all real numbers" is true.

Step5: Analyze the range of \( g(x) \)

Since the domain is all real numbers and the function \( g(x)=-\frac{1}{2}x^3 \) is a cubic function (odd - degree polynomial), it is a one - to - one function that can take on any real number value. For example, for any real number \( y \), we can solve \( y=-\frac{1}{2}x^3 \) for \( x \) as \( x =-\sqrt[3]{2y} \), which is a real number for any real \( y \). So the range is all real numbers. Thus, "The range of the function is all real numbers" is true.

Answer:

• As \( x \) approaches negative 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.