Problem 109 - Precalculus 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 graph of \(f(x) = \sqrt[3]{x}\) shifted left by 2 units.

Step 1: Graph the basic function

Start by graphing the cube root function \(f(x)=\sqrt[3]{x}\). This function is characterized by its turning point at the origin (0, 0), and it increases as \(x\) increases.

Step 2: Understand the transformation

The function \(g(x)=\sqrt[3]{x+2}\) is a transformation of the original function. The '+2' inside the cube root shifts the graph horizontally to the left by 2 units.

Step 3: Graph transformed function

Draw the graph of \(g(x)=\sqrt[3]{x+2}\) by shifting each point on the graph of \(f(x) = \sqrt[3]{x}\) to the left by 2 units. This is the graph of transformed function.