Question

graph the function and give its domain and range. \\( f(x) = \sqrt3{x - 1} \\) choose the correct graph. \\( \bigcirc \\) a. graph \\( \bigcirc \\) b. graph \\( \bigcirc \\) c. graph \\( \bigcirc \\) d. graph

Explanation:

Step1: Analyze the function \( f(x)=\sqrt[3]{x - 1} \)

The parent function of a cube root function is \( y = \sqrt[3]{x} \), which has a domain of all real numbers (\( (-\infty, \infty) \)) and a range of all real numbers (\( (-\infty, \infty) \)). The function \( f(x)=\sqrt[3]{x - 1} \) is a horizontal shift of the parent cube root function 1 unit to the right. The shape of the cube root function is an S - shaped curve. For a cube root function \( y=\sqrt[3]{u} \), where \( u=x - 1 \), since we can take the cube root of any real number (positive, negative, or zero), the domain of \( f(x) \) is all real numbers, and the range is also all real numbers.

Step2: Analyze the graphs

• Option A: The graph has the characteristic S - shape of a cube root function, shifted 1 unit to the right (we can check key points: when \( x = 1 \), \( f(1)=\sqrt[3]{1 - 1}=0 \); when \( x=2 \), \( f(2)=\sqrt[3]{2 - 1}=1 \); when \( x = 0 \), \( f(0)=\sqrt[3]{0 - 1}=- 1 \)). The graph should pass through \((1,0)\), \((2,1)\), and \((0, - 1)\), which is consistent with the shape of a cube root function.
• Option B: The graph looks like an exponential decay or a logarithmic function (not the S - shaped curve of a cube root function), so it can be eliminated.
• Option C: The graph does not match the key points or the shape of the cube root function shifted 1 unit to the right.
• Option D: The graph also does not have the S - shape of a cube root function, more like a horizontal line or a very flat curve, so it can be eliminated.

Answer:

A. (the graph corresponding to option A)