Problem 110 - 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 graph of the cube root function, shifted 2 units to the right.

Step 1: Graph of Original Function

Begin by drawing the graph of the cube root function \(f(x)=\sqrt[3]{x}\). This function's graph is a curve that opens upward, passing through the origin. When \(x>0\), it is increasing, and when \(x<0\), it is decreasing.

Step 2: Transformation Rules

When the function inside the root symbol is transformed from \(x\) to \(x-c\) where \(c\) is positive, the graph of the function moves to the right by \(c\) units. This is a horizontal shift.

Step 3: Graph of Transformed Function

Given \(g(x)=\sqrt[3]{x-2}\), this is the cube root function shifted right by 2 units. So, take every point on original cube root function and move it 2 units to the right to get the graph of \(g(x)\). The graph still has the same shape as before, but each point's x-coordinate is 2 units larger.