Question

identify the graph of \\(y = x^3\\).

the domain of \\(y = x^3\\) is
all real numbers.
\\(x > 0\\).
\\(x \ge 0\\).

the range of \\(y = x^3\\) is
all real numbers.
\\(y > 0\\).
\\(y \ge 0\\).

Explanation:

Identify the graph of \(y = x^3\)

Using the Cubic Function Analysis knowledge point

$$
LATEXBLOCK0
$$

The graph must pass through \((-1, -1)\), \((0, 0)\), and \((1, 1)\), which corresponds to the second graph.

Determine the domain of \(y = x^3\)

Using the Domain of Polynomial Functions knowledge point

$$ \text{Domain} = (-\infty, \infty) $$

Since \(y = x^3\) is a polynomial function, it is defined for all real numbers \(x\).

Determine the range of \(y = x^3\)

To find the range of the cubic parent function \(y = x^3\), we analyze its behavior as \(x\) approaches positive and negative infinity.
As \(x \to \infty\), \(y \to \infty\).
As \(x \to -\infty\), \(y \to -\infty\).
Since the function is continuous and strictly increasing, it takes on every real value.
Therefore, the range is all real numbers.

Answer:

Question 1

• (A) Exponential-like curve passing through (0,1)
• (B) S-shaped cubic curve passing through (-1,-1), (0,0), and (1,1) (Correct answer)

Question 2

• all real numbers. (Correct answer)
• x > 0.
• x \(\ge\) 0.

Question 3

• all real numbers. (Correct answer)
• y > 0.
• y \(\ge\) 0.