Question

cubic & cube root functions unit online f
complete this assessment to review what you’ve learned. it
what is the range of $g(x) = -sqrt3{x + 5}$? (1 point)
$\bigcirc\\ -infty, infty$\
$\bigcirc\\ (-infty, 0$\
$\bigcirc\\ 0, infty)$\
$\bigcirc\\ (-infty, infty)$

Explanation:

Step1: Recall cube root function range

The parent cube root function \( y = \sqrt[3]{x} \) has a range of \( (-\infty, \infty) \) because for any real number \( y \), we can find an \( x \) (specifically \( x = y^3 \)) such that \( \sqrt[3]{x}=y \).

Step2: Analyze transformations

The function \( g(x)=-\sqrt[3]{x + 5} \) is a transformation of the parent cube root function. The horizontal shift \( x+5 \) (replacing \( x \) with \( x + 5 \)) does not affect the range (it only shifts the graph left/right). The negative sign in front is a reflection over the \( x \)-axis, but reflection over the \( x \)-axis of a function with range \( (-\infty, \infty) \) still results in a range of \( (-\infty, \infty) \), because if \( y \) can be any real number for \( \sqrt[3]{x} \), then \( -y \) can also be any real number.

Answer:

\( (-\infty, \infty) \) (the option with \( (-\infty, \infty) \))