Problem 117 - 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 \(g(x) = \sqrt[3]{-x-2}\) is the graph of \(f(x) = \sqrt[3]{x}\) that's been flipped horizontally and shifted two units to the right.

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

Plot the cube root function by choosing a variety of x-values, finding their cube roots, and then plotting the coordinates on a graph. Cube roots of negative numbers are negative and those of positive numbers are positive. So, the graph will go through the points (-1,-1), (0,0), and (1,1), giving a curve that increases slowly, crossing the x and y axes at the origin.

Step 2: Understand The Transformation

The function \(g(x)=\sqrt[3]{-x-2}\) is the base cube root function but flipped horizontally (because of the negative sign before x) and shifted 2 units to the right (because of -2 inside the cube root).

Step 3: Apply The Transformation and Graph

Flip the graph of \(f(x) = \sqrt[3]{x}\) over the y-axis to account for the negative sign before x. Then, shift the flipped graph 2 units to the right to account for the -2 inside the cube root to get the graph for \(g(x)\). It helps to take some positive and negative x values, evaluate their function values and plot these points to ensure accuracy.