Problem 118 - 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 the function \(g(x)=\sqrt[3]{-x+2}\) is the graph of the cube root function \(f(x)=\sqrt[3]{x}\) reflected over the y-axis and then shifted left by 2 units.

Step 1: Graph the Cube Root Function

Begin by graphing the cube root function \(f(x)=\sqrt[3]{x}\). This function is the root function, and its graph starts from the origin (0,0) and grows in the direction of both the positive and negative y-axis as x increases or decreases respectively.

Step 2: Identify the Transformations

Upon comparing the function \(g(x)=\sqrt[3]{-x+2}\) with \(f(x)=\sqrt[3]{x}\), it can be seen that there are two transformations: a reflection in the y-axis due to the negative sign before x, and a horizontal shift to the left by 2 units due to the +2 inside the cube root.

Step 3: Apply the Transformations and Graph

Apply the identified transformations to the initial graph. For a reflection in the y-axis, the graph of \(f(x)\) will be mirrored along the y-axis. And for a horizontal shift left by 2 units, every point on the graph of \(f(x)\) will be moved 2 units to the left. Use these transformations to draw \(g(x)=\sqrt[3]{-x+2}\).