Problem 110 - 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 the function \(g(x) = \sqrt[3]{x - 2}\) can be obtained by taking the graph of the function \(f(x) = \sqrt[3]{x}\), and shifting it 2 units to the right. This means the function \(g(x)\) will be undefined for values of x less than 2, and will cross the x-axis at (2,0).

Step 1: Graph the Cube Root Function, \(f(x) = \sqrt[3]{x}\)

Begin by sketching the cube root function \(f(x) = \sqrt[3]{x}\). This function rises slowly, crosses the origin at (0,0), and continues to rise in the positive direction. The function is undefined for negative values of x.

Step 2: Understand the Transformation

Looking at the transformed function \(g(x) = \sqrt[3]{x - 2}\), it is evident that every \(x\) in the original function \(f(x) = \sqrt[3]{x}\) has been replaced by \((x - 2)\). This represents a horizontal translation (or shift) of 2 units to the right.

Step 3: Produce the Graph of the Transformed Function

Draw the graph of function \(g(x) = \sqrt[3]{x - 2}\) by shifting the graph of function \(f(x) = \sqrt[3]{x}\) two units to the right. This means, every point \((x, y)\) on the graph of function \(f(x)\) will move to the point \((x+2, y)\) on the graph of function \(g(x)\). So, instead of crossing the origin, the graph now crosses through (2,0)