Problem 58 - A Graphical Approach to Precalculus with Limits

Question

For each function that is one-to-one, write an equation for the inverse function of \(y=f(x)\) in the form \(y=f^{-1}(x),\) and then graph \(f\) and \(f^{-1}\) on the same axes. Give the domain and range of \(f\) and \(f^{-1} .\) If the function is not one-to-one, say so. $$y=-x^{3}-2$$

Step-by-Step Solution

Answer

The inverse is \( y = \sqrt[3]{-x - 2} \). Domain and range for both \( f \) and \( f^{-1} \) are all real numbers.

Step 1: Check if the function is one-to-one

A function is one-to-one if it passes the horizontal line test: any horizontal line should not intersect the graph of the function more than once. The function \( y = -x^3 - 2 \) is a cubic function, which is typically one-to-one as it is strictly increasing or decreasing. Hence, this function is one-to-one.

Step 2: Find the inverse function

To find the inverse, start by replacing \( y \) with \( x \) and solve for \( y \). Start with \( x = -y^3 - 2 \). Then: 1. Add 2 to both sides: \( x + 2 = -y^3 \).2. Multiply both sides by -1: \( -x - 2 = y^3 \).3. Take the cube root of both sides: \( y = \sqrt[3]{-x - 2} \).The inverse function is \( y = f^{-1}(x) = \sqrt[3]{-x - 2} \).

Step 3: Graph the functions

Graph the original function \( y = -x^3 - 2 \). It is a cubic graph, shifted downward by 2 units from \( y = -x^3 \). Next, graph the inverse function \( y = \sqrt[3]{-x - 2} \), which is essentially a reflection of the original function across the line \( y = x \) due to the relationship between a function and its inverse. Use a graphing tool or software to accurately plot both functions on the same set of axes for better visual comparison.

Step 4: Determine the domain and range

For \( y = -x^3 - 2 \), the domain is all real numbers \((-fty, fty)\) because cubic functions can take any real number as input. The range is also all real numbers \((-fty, fty)\) because it returns every real number given enough input.For \( y = f^{-1}(x) = \sqrt[3]{-x - 2} \), both the domain and range are also all real numbers \((-fty, fty)\). This is typical for cube root functions, which can receive any real number and produce any real number.

Key Concepts

Understanding One-to-One Functions

When dealing with inverse functions, determining if a function is one-to-one is crucial. A function is categorized as one-to-one if it passes the horizontal line test. This means that any horizontal line drawn across the graph of the function intersects it at most once. Such behavior guarantees that every unique output (or \'y\' value) is produced by exactly one unique input (or \'x\' value).

Let's consider the cubic function example given by the exercise: \( y = -x^3 - 2 \). Cubic functions like this one are usually one-to-one because they steadily increase or decrease, never changing direction.

• Strictly increasing or decreasing functions pass the horizontal line test.
• If a function is not one-to-one, finding its inverse would not result in a function.

Recognizing one-to-one functions is essential when seeking their inverses, as it ensures each output is unique to a single input, maintaining a valid function relationship.

Domain and Range

The domain and range provide crucial insights into a function's behavior. The domain refers to all possible \'x\' values that a function can accept, while the range consists of all possible \'y\' values that the function can produce.

For the cubic function \( y = -x^3 - 2 \):

• The domain consists of all real numbers \((-\infty, \infty)\) because cubic functions accept all real inputs.
• Similarly, the range is all real numbers, as the function can produce any real output over its domain.

When considering inverse functions, such as \( y = \sqrt[3]{-x - 2} \), understanding domain and range helps reflect how the function maps inputs to outputs.

• The inverses typically mirror the domain and range of the original function.
• This cube root function also behaves over all real numbers, both for domain and range.

Keeping domain and range in mind proves valuable, especially when translating these concepts to an inverse function.

Graphing Functions and Their Inverses

Visualizing functions is a powerful tool in understanding relationships, particularly between a function and its inverse. When graphing, remember that a function's inverse is a reflection across the line \( y = x \).

Consider our function \( y = -x^3 - 2 \). This graph shifts the standard cubic graph downward by 2 units. By graphing it and its inverse, \( y = \sqrt[3]{-x - 2} \), you reveal their relationship more clearly.

• The line \( y = x \) acts as a mirror for function and inverse.
• Noticing symmetry can clarify why inverse functions have their specific domains and ranges.
• Graphing aids in confirming the one-to-one status of functions visually.

Try using graphing tools or software to actively compare and highlight these reflections, reinforcing your understanding of function inverses.