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 is a real number\\} \bigcirc \\{y \mid y is a real number\\}

Explanation:

Step1: Recall domain of cube root function

The parent function \( y = \sqrt[3]{x} \) has a domain of all real numbers because we can take the cube root of any real number (positive, negative, or zero).

Step2: Analyze the translation

The given function is \( y = \sqrt[3]{x - 1} + 3 \). Translations (horizontal shift here by 1 unit right, vertical shift by 3 units up) do not affect the domain of a cube root function. The expression inside the cube root, \( x - 1 \), is defined for all real numbers \( x \) (since we can subtract 1 from any real number and still have a real number, and then take its cube root). So the domain remains all real numbers for \( x \).

Step3: Evaluate the options

• Option 1: \( \{x\mid 1 < x < 5\} \) is incorrect as the graph extends beyond this interval.
• Option 2: \( \{y\mid 1 < y < 5\} \) is about the range (y - values), not domain (x - values), so incorrect.
• Option 3: \( \{x\mid x \text{ is a real number}\} \) matches our conclusion about the domain (x - values can be any real number).
• Option 4: \( \{y\mid y \text{ is a real number}\} \) is about the range, not domain, so incorrect.

Answer:

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