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 \ge -4 \\)
\\( g(x) = \sqrt3{x} + 4; x \ge 0 \\)
\\( g(x) = \sqrt3{x + 4}; x \ge -4 \\)
\\( g(x) = \sqrt3{x} - 4; x \ge 0 \\)

Explanation:

Step1: Analyze Domain from Graph

The graph starts at \( x = -4 \) (since 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 Function Form

The parent cube - root function is \( y=\sqrt[3]{x} \). A horizontal shift of \( h \) units is given by \( y = \sqrt[3]{x - h} \) (shift right \( h \)) or \( y=\sqrt[3]{x+h}=\sqrt[3]{x-(-h)} \) (shift left \( h \)). The graph here is a left - shifted or right - shifted cube - root function? Let's check the y - intercept. When \( x = 0 \), let's see the value. For the third option \( g(x)=\sqrt[3]{x + 4} \), when \( x = 0 \), \( g(0)=\sqrt[3]{0 + 4}=\sqrt[3]{4}\approx1.59 \), but from the graph, when \( x = 0 \), \( y\approx2 \)? Wait, no, let's check the starting point. The starting point is at \( x=-4,y = 0 \)? Wait, no, the graph at \( x=-4 \), what's \( y \)? Wait, the graph starts at \( x=-4 \), and when \( x=-4 \), \( \sqrt[3]{-4 + 4}=\sqrt[3]{0}=0 \)? Wait, no, the graph at \( x = - 4 \) seems to have \( y = 0 \)? Wait, no, looking at the graph, when \( x=-4 \), the function starts, and as \( x \) increases, \( y \) increases. Let's check 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 does not match the graph (the graph at \( x=-4 \) is at \( y = 0 \) or near \( y = 0 \)? Wait, no, maybe I misread the graph. Wait, the third option: \( g(x)=\sqrt[3]{x + 4} \), domain \( x\geq - 4 \). When \( x=-4 \), \( g(-4)=\sqrt[3]{-4 + 4}=0 \), which matches the left - most point (at \( x=-4 \), the function starts, probably \( y = 0 \) there). Then, as \( x \) increases, \( \sqrt[3]{x + 4} \) increases. Let's check the y - intercept (\( x = 0 \)): \( g(0)=\sqrt[3]{0 + 4}=\sqrt[3]{4}\approx1.59 \), but the graph at \( x = 0 \) is around \( y = 2 \)? Wait, no, maybe the graph is \( g(x)=\sqrt[3]{x + 4} \). Wait, the first option: \( g(x)=\sqrt[3]{x - 4} \), domain \( x\geq - 4 \). When \( x=-4 \), \( g(-4)=\sqrt[3]{-8}=-2 \), which is not matching the graph (the graph at \( x=-4 \) is above the x - axis? Wait, no, the graph in the picture: the leftmost point is at \( x=-4 \), and the y - value there is 0? Wait, the x - axis is crossed at \( x=-4 \)? Wait, the graph starts at \( x=-4 \), and goes up. So for \( g(x)=\sqrt[3]{x + 4} \), when \( x=-4 \), \( y = 0 \), and as \( x \) increases, \( y=\sqrt[3]{x + 4} \) increases, which matches the shape of the cube - root function (since the cube - root function is increasing). The first option \( g(x)=\sqrt[3]{x - 4} \) would have a right shift, and at \( x=-4 \), \( y=\sqrt[3]{-8}=-2 \), which does not match the graph. So the correct function is \( g(x)=\sqrt[3]{x + 4},x\geq - 4 \).

Answer:

\( g(x)=\sqrt[3]{x + 4};x\geq - 4 \) (the third option: \( g(x)=\sqrt[3]{x + 4};x\geq - 4 \))