Question

the graph shows the function g(x) for a restricted domain.
which is the function g(x) for a restricted domain?
\\( g(x) = \sqrt3{x - 4}; x \geq -4 \\)
\\( g(x) = \sqrt3{x} + 4; x \geq 0 \\)
\\( g(x) = \sqrt3{x + 4}; x \geq -4 \\)
\\( g(x) = \sqrt3{x} - 4; x \geq 0 \\)

Explanation:

Step1: Analyze the domain

The graph starts at \( x = -4 \) (the leftmost point is at \( x = -4 \)), so the domain is \( x \geq -4 \). This eliminates options with \( x \geq 0 \) (second and fourth options).

Step2: Analyze the function form

The parent function of the cube root is \( y=\sqrt[3]{x} \). A horizontal shift: \( y = \sqrt[3]{x - h} \) shifts right \( h \), \( y=\sqrt[3]{x + h} \) shifts left \( h \). The graph here seems to be a cube root function shifted left 4 units (since it starts at \( x=-4 \), which is a shift of the parent cube root function \( y = \sqrt[3]{x} \) (which has domain all real numbers, but here restricted) or a shift. Wait, the general form for a cube root function with horizontal shift: if the vertex (the point where the curve starts) is at \( x=-4 \), let's check the function. Let's take the third option: \( g(x)=\sqrt[3]{x + 4} \), when \( x=-4 \), \( g(-4)=\sqrt[3]{-4 + 4}=0 \)? Wait no, the graph at \( x=-4 \), what's the y - value? Wait the graph at \( x=-4 \), the y - value is around 0? Wait no, looking at the graph, at \( x=-4 \), the curve starts, and when \( x = 0 \), let's see the y - value. Let's check the third option: \( g(x)=\sqrt[3]{x + 4} \), domain \( x\geq - 4 \). When \( x=-4 \), \( g(-4)=\sqrt[3]{-4 + 4}=0 \)? Wait the graph at \( x=-4 \), the y - coordinate: looking at the grid, at \( x=-4 \), the point is ( - 4, 0)? Wait no, the graph starts at \( x=-4 \), and at \( x = 0 \), let's see the y - value. Let's check the third option: \( g(0)=\sqrt[3]{0 + 4}=\sqrt[3]{4}\approx1.587 \), but the graph at \( x = 0 \) has y - value around 1.5? Wait maybe. Wait the first option: \( g(x)=\sqrt[3]{x - 4} \), domain \( x\geq - 4 \). When \( x=-4 \), \( g(-4)=\sqrt[3]{-4 - 4}=\sqrt[3]{-8}=-2 \), which doesn't match the graph (the graph at \( x=-4 \) is at y = 0 or near 0). So first option is out. Third option: \( g(x)=\sqrt[3]{x + 4} \), domain \( x\geq - 4 \). When \( x=-4 \), \( g(-4)=0 \), and as \( x \) increases, \( g(x) \) increases, which matches the graph's shape (a cube root curve, increasing, starting at \( x=-4 \)).

Step3: Confirm the domain and function

We already saw that the domain is \( x\geq - 4 \) (eliminating options with \( x\geq0 \)). Then between the first and third options: first option is \( g(x)=\sqrt[3]{x - 4} \), domain \( x\geq - 4 \). When \( x=-4 \), \( g(-4)=\sqrt[3]{-8}=-2 \), which doesn't match the graph (the graph at \( x=-4 \) has y - value 0 or near 0). The third option: \( g(x)=\sqrt[3]{x + 4} \), domain \( x\geq - 4 \), when \( x=-4 \), \( g(-4)=0 \), and as \( x \) increases, the function increases, which matches the graph's behavior.

Answer:

C. \( g(x)=\sqrt[3]{x + 4};x\geq - 4 \) (Note: In the options, the third option is \( g(x)=\sqrt[3]{x + 4};x\geq - 4 \))