As a subject matter expert in mathematics, I specialize in the study of functions and their properties. Understanding the range of a function is crucial in mathematics, as it helps us predict the possible outputs for any given input. In the context of the cubic parent function, we're talking about the simplest form of a cubic function, which is \( f(x) = x^3 \).
The domain of a function is the set of all real values of \( x \) that will give real values for \( y \). In the case of the cubic parent function \( f(x) = x^3 \), the domain is all real numbers, because no real number raised to the third power is an imaginary number.

The range of a function is the set of all real values of \( y \) that you can get by plugging real numbers into \( x \). For the cubic parent function \( f(x) = x^3 \), the range is also all real numbers. This is because no matter what real number you choose for \( x \), the result of \( x^3 \) will always be a real number.

To visualize this, consider the graph of the cubic parent function. It is a smooth curve that passes through the origin (0,0) and extends infinitely in both the positive and negative directions along the \( y \)-axis. This is because as \( x \) approaches positive or negative infinity, \( x^3 \) also approaches positive or negative infinity, respectively. Conversely, when \( x \) is 0, \( x^3 \) is also 0.

The cubic parent function is an odd function, which means that \( f(-x) = -f(x) \). This property is evident in its graph, which is symmetric with respect to the origin. This symmetry also contributes to the fact that the range includes all real numbers, as for every positive \( y \) value, there is a corresponding negative \( y \) value.

In summary, the range of the cubic parent function \( f(x) = x^3 \) is all real numbers, denoted as \( (-\infty, \infty) \). This is a fundamental characteristic of cubic functions, and understanding it is key to analyzing more complex cubic functions and their behavior.

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The domain of a function is the set of all real values of x that will give real values for y. The range of a function is the set of all real values of y that you can get by plugging real numbers into x. The quadratic parent function is y = x2. The graph of this function is shown below.read more >>