Problem 108 - 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

First, the cube root function \(f(x)=\sqrt[3]{x}\) is plotted. This function is positively increasing for positive \(x\) and negatively increasing for negative \(x\), crossing the x-axis at origin. To plot the given function \(g(x)=\sqrt[3]{x}-2\), every point on the graph of \(f(x)\) is shifted vertically down by 2 units. The result is a graph similar to \(f(x)\) but situated 2 units lower on the y-axis.

Step 1 Graph the cube root function \(f(x)\)

Begin by plotting the function \(f(x)=\sqrt[3]{x}\). The cube root function is defined for all real numbers, and it crosses the x-axis at the origin (0,0). As \(x\) moves towards the positive side, the function slowly rises, while as \(x\) moves towards the negative side, the function slowly decreases.

Step 2 Recognize the transformation

Note that the original function \(f(x)\) has been transformed to \(g(x) = f(x) - 2\), which represents a vertical shift downwards by 2 units. In other words, every y-coordinate of \(f(x)\) is reduced by 2 to create the graph for \(g(x)\).

Step 3 Plot the transformed function \(g(x)\)

Use the graph of \(f(x)\) to plot \(g(x)\). For every point on \(f(x)\), move it down two units to find the corresponding point on \(g(x)\). The graph still crosses the x-axis in the same places because the transformation only moves the graph vertically, not horizontally. Plot these points to create the graph of \(g(x)=\sqrt[3]{x}-2\).