Question

on the coordinate grid, the graph of $y = \sqrt3{x - 1} + 3$ is shown. it is a translation of $y = \sqrt3{x}$. what is the domain of the graphed function? \bigcirc \\{x \mid 1 < x < 5\\} \bigcirc \\{y \mid 1 < y < 5\\} \bigcirc \\{x \mid x \text{ is a real number}\\}\bigcirc \\{y \mid y \text{ is a real number}\\}

Explanation:

Step1: Recall Domain Definition

Domain is all possible \( x \)-values of a function. For cube root functions \( y = \sqrt[3]{u} \), the expression inside the cube root (\( u \)) can be any real number (since cube root of negative, zero, positive reals is defined).

Step2: Analyze the Given Function

The function is \( y = \sqrt[3]{x - 1} + 3 \). The inside of the cube root is \( x - 1 \). Since cube root is defined for all real \( x - 1 \) (i.e., all real \( x \)), there are no restrictions on \( x \).

Step3: Evaluate Options

• Option 1: \( \{x | 1 < x < 5\} \) is incorrect (no restriction to \( 1 < x < 5 \)).
• Option 2: \( \{y | 1 < y < 5\} \) is range (y-values), not domain.
• Option 3: \( \{x | x \text{ is a real number}\} \) matches, as \( x \) can be any real.
• Option 4: \( \{y | y \text{ is a real number}\} \) is range, not domain.

Answer:

\(\{x | x \text{ is a real number}\}\) (the third option)