Problem 108 - 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 similar to the graph of the base function \(f(x)=\sqrt[3]{x}\) but shifted down by 2 units.

Step 1: Understanding the base function

The base function is the cube root function \(f(x)=\sqrt[3]{x}\). It's a function that for any real number, gives its cube root. It's essential to understand how this function looks like. The cube root of x function is an odd function, which means it's symmetric about the origin. The graph passes through the origin (0, 0), and as x increases or decreases, so does \(f(x)\).

Step 2: Graphing the base function

Make a simple sketch of the cube root function. You can do this by plotting some easy points. For instance, for \(x=-8, -1, 0, 1, 8\), the cube root function takes the values \(-2, -1, 0, 1, 2\) respectively. Plot these points and connect them with a smooth curve, indicating the general shape of the function, which is symmetric about the origin.

Step 3: Understanding the transformation

Given the function \(g(x)=\sqrt[3]{x}-2\), it's a transformation of the base function by moving every point on it down by 2 units. This is because the '-2' in the function expressing subtracting 2 units from every output of the base function.

Step 4: Applying the Transformation

Translate every point on the sketch of \(f(x)=\sqrt[3]{x}\) downward by 2 units. This new function, \(g(x)=\sqrt[3]{x}-2\), will have the same general shape and symmetry as the cube root function, but it is shifted down. For instance, the point on \(f(x)\) that was at (8, 2) will now be at (8, 0) on \(g(x)\), and the point that was at (-8, -2) will now be at (-8, -4) on \(g(x)\).