Problem 73 - A Graphical Approach to College Algebra

Question

Functions such as the pairs in Exercises \(69-72\) are called inverse functions, because the result of composition in both directions is the identity function. (Inverse functions will be discussed in detail in Section 5.1.) In a square viewing window, graph \(y_{1}=\sqrt[3]{x-6}\) and \(y_{2}=x^{3}+6,\) an example of a pair of inverse functions. Now graph \(y_{3}=x .\) Describe how the graph of \(y_{2}\) can be obtained from the graph of \(y_{1},\) using the graph \(y_{3}=x\) as a basis for your description.

Step-by-Step Solution

Answer

Reflecting \(y_1\) across \(y_3 = x\) gives the graph of \(y_2\).

Step 1: Identify the inverse functions

First, we identify the functions given: \(y_1 = \sqrt[3]{x-6}\) and \(y_2 = x^3 + 6\). These are inverse functions because their compositions result in the identity function: \(y_1(y_2) = x\) and \(y_2(y_1) = x\). This means that \(y_1(y_2(x)) = x\) and vice versa.

Step 2: Graph the functions

In a square viewing window, plot the function \(y_1 = \sqrt[3]{x-6}\), which is a cubic root function, horizontally shifted to the right by 6 units. Next, plot \(y_2 = x^3 + 6\), which is a cubic function, vertically shifted up by 6 units. Finally, plot \(y_3 = x\), the identity function which passes through the origin, making a 45-degree line with the x-axis.

Step 3: Describe how to obtain graph of y2 from y1 using y3 as basis

The graph \(y_2 = x^3 + 6\) can be obtained from \(y_1 = \sqrt[3]{x-6}\) by reflecting \(y_1\) across the line \(y_3 = x\). Inverse functions are symmetric around the line \(y = x\). Therefore, graphing \(y_2\) involves flipping the orientation of \(y_1\) over the \(y = x\) line.

Key Concepts

Identity Function

When learning about inverse functions, one key concept to grasp is the identity function. The identity function is simply represented as \( y = x \), where every input value is equal to its output. This means, for example, if you input 2, the output will also be 2. The identity function is often used as a test to check if two functions are inverses of each other.

The reason it's so important is that inverse functions, when composed, give you back this function. For instance, if you have two functions, \( y_1 \) and \( y_2 \), and they are inverses, then \( y_1(y_2(x)) = x \) and \( y_2(y_1(x)) = x \). Therefore, they effectively "cancel" each other out, returning the original input. This concept is foundational for understanding how inverses work and why they are useful in mathematics. Let this characteristic be your guide for verifying inverses.

Graphing Functions

Graphing functions is a fundamental skill in understanding their behavior and relationships. Start with the functions given, such as \( y_1 = \sqrt[3]{x-6} \) and \( y_2 = x^3 + 6 \), by identifying their forms.

The function \( y_1 = \sqrt[3]{x-6} \) is a cubic root function. In this case, it's shifted horizontally to the right by 6 units. Imagine how the graph moves from the origin along the x-axis. Meanwhile, \( y_2 = x^3 + 6 \) is a classic cubic function shifted vertically upwards by 6 units. This shift means that every point on the basic cubic graph moves 6 units higher in the y-direction.

Lastly, the line \( y_3 = x \) is the identity function. It's a simple, straightforward line that runs diagonally at 45 degrees through the origin. This graph is your reference point for analyzing symmetry and relationships between \( y_1 \) and \( y_2 \). Each graph reveals different shifts and transformations, key in analyzing inverse functions.

Function Composition

Function composition involves creating a new function by combining two functions, say, \( f \) and \( g \), such that the output of \( g \) becomes the input of \( f \). This is denoted as \( f(g(x)) \).

In the context of inverse functions, as shown by the functions \( y_1 \) and \( y_2 \) in the exercise, composition is significant for determining if two functions are inverses. When you perform the composition \( y_1(y_2(x)) \), you should end up with \( x \) if these functions are true inverses. This composition reflects the undoing each function does to the other's effect, essentially bringing us back to the starting point.

• Inverse composition: flips function actions
• Symmetry: function graph reflects over line \( y = x \)

Understanding this concept clarifies how two functions interact and simplifies complex function operations.