Question

graphing the cubic function quick check
given the function $f(x) = 5x^3 - 7$, which of the following describes its domain?
$\circ$ $(-\infty, 7)$
$\circ$ $(-\infty, \infty)$
$\circ$ $(-7, 7)$
$\circ$ $(7, \infty)$

Explanation:

Step1: Recall Domain of Cubic Functions

A cubic function is of the form \( f(x) = ax^3 + bx^2 + cx + d \) (or simpler forms like \( 5x^3 - 7 \)). For polynomial functions (including cubic functions), there are no restrictions on the input \( x \) (no division by zero, no square roots of negative numbers here). So the domain of any polynomial function is all real numbers.

Step2: Analyze the Given Function

The function \( f(x) = 5x^3 - 7 \) is a cubic polynomial. Since it's a polynomial, \( x \) can take any real value (positive, negative, zero). So the domain is \( (-\infty, \infty) \).

Answer:

\( (-\infty, \infty) \) (the option with this interval, e.g., if the options are labeled as above, the second option: \( (-\infty, \infty) \))