Question

on the coordinate grid, the graph of $y = \sqrt3{-x} - 1$ is shown. it is a reflection and translation of $y = \sqrt3{x}$. what is the range of the graphed function? \bigcirc \\ \\{x \mid -2 < x < 2\\} \bigcirc \\ \\{y \mid -2 < y < 2\\} \bigcirc \\ \\{x \mid x \text{ is a real number}\\}\bigcirc \\ \\{y \mid y \text{ is a real number}\\}

Explanation:

Step1: Recall the range of cube root function

The parent function \( y = \sqrt[3]{x} \) has a range of all real numbers 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 \( y=\sqrt[3]{-x - 1}\) is a reflection and translation of \( y = \sqrt[3]{x}\). Reflections and translations (horizontal/vertical shifts, reflections over axes) of the cube - root function do not change the fact that the range remains all real numbers. The range of a function is the set of all possible \( y \) - values. So we are looking for the set of \( y \) - values, not \( x \) - values. Options that involve \( x \) (like \(\{x|-2 < x < 2\}\) and \(\{x|x\text{ is a real number}\}\)) are incorrect because they describe the domain (set of \( x \) - values) or an incorrect description of the range. The option \(\{y|-2 < y < 2\}\) is incorrect because the graph of the cube - root function (and its transformations) extends infinitely in the vertical direction, so \( y \) can take on any real number value.

Answer:

\(\{y\mid y\text{ is a real number}\}\)