Problem 109 - College Algebra Essentials

Question

Begin by graphing the cube root function, \(f(x)=\sqrt[3]{x} .\) Then use transformations of this graph to graph the given function. $$g(x)=\sqrt[3]{x+2}$$

Step-by-Step Solution

Answer

The graph of \(g(x) = \sqrt[3]{x+2}\) is the same as the graph of the cube root function \(f(x) = \sqrt[3]{x}\), but shifted 2 units to the left.

Step 1: Graph the cube root function

Sketch the graph of \(f(x)=\sqrt[3]{x}\). It should look roughly as follows: The graph passes through the origin \((0,0)\), and for any point \((a, a^3)\), there is a point \((a, -a^3)\) on the graph.

Step 2: Understand the transformation

The graph of \(g(x)=\sqrt[3]{x+2}\) is the graph of \(f(x)=\sqrt[3]{x}\), shifted 2 units to the left. This is because adding 2 inside the function's argument offsets the function's graph to the left by 2 units.

Step 3: Apply the transformation to graph \(g(x)\)

Shift every point in the graph of \(f(x)\) 2 units to the left to create the graph of \(g(x) = \sqrt[3]{x+2}\). The cube root function typically looks the same no matter how you shift it around the coordinate grid, so the graph of \(g(x)\) should look similar to the initial graph of \(f(x)\) but shifted 2 units to the left.