Problem 48 - College Algebra

Question

For the following exercises, use a calculator to graph the function. Then, using the graph, give three points on the graph of the inverse with \(y\) -coordinates given. $$f(x)=x^{3}+3 x-4, y=0,1,2$$

Step-by-Step Solution

Answer

The inverse points are approximately (0,1.35), (1,1.41), and (2,1.51).

Step 1: Graph the Function

First, graph the function \( f(x) = x^3 + 3x - 4 \) using a graphing calculator. This image will help us visualize the behavior of the function. Note how the curve progresses through various regions on the Cartesian plane.

Step 2: Understand the Inverse Relationship

Remember that if \((a, b)\) is a point on the graph \( y = f(x) \) of a function, then \((b, a)\) is a point on the graph of its inverse. We will find specific points on the inverse graph given particular \( y \)-coordinates.

Step 3: Identify Points on the Graph

For each given \( y \)-coordinate (0, 1, and 2), identify the corresponding \( x \)-coordinates by solving \( f(x) = y \). These \( x \) values form the \( y \)-coordinates of points on the graph of the inverse.

Step 4: Find Points for \( y = 0 \)

Set the equation \( f(x) = 0 \), which simplifies to \( x^3 + 3x - 4 = 0 \). Solve for \( x \). The solution will give us a point \((0, x_1)\) for the inverse graph.

Step 5: Find Points for \( y = 1 \)

Set the equation \( f(x) = 1 \), which simplifies to \( x^3 + 3x - 5 = 0 \). Solve for \( x \). The solution will give us a point \((1, x_2)\) for the inverse graph.

Step 6: Find Points for \( y = 2 \)

Set the equation \( f(x) = 2 \), which simplifies to \( x^3 + 3x - 6 = 0 \). Solve for \( x \). The solution will give us a point \((2, x_3)\) for the inverse graph.

Key Concepts

Graphing Calculators

Graphing calculators are powerful tools to visualize mathematical functions quickly. When tackling problems like graphing functions and their inverses, a graphing calculator becomes essential. Its ability to plot a graph electronically allows you to see the function's behavior without manually plotting each point.
Using a graphing calculator involves:

• Inputting the function: Simply enter the function such as \(f(x) = x^3 + 3x - 4\) into the calculator.
• Viewing the graph: The calculator displays how the curve looks, and you can use this to understand key features of the function like intercepts and general shape.
• Zoom tools: These features allow you to zoom in on specific parts of the graph for detailed analysis.

These steps help ensure a clearer understanding of the function, making it easier to find inverse points later.

Function Analysis

Analyzing a function is the key to understanding its properties. Each function has unique characteristics based on its equation. For the function \(f(x) = x^3 + 3x - 4\), let's look deeper.

• Behavior: This function is a cubic polynomial. Cubic functions generally have an "S" shaped curve and continue infinitely in both vertical directions.
• Intercepts: Functions are often analyzed by finding the x-intercepts (where \(f(x) = 0\)) and y-intercepts (where \(x = 0\)). Knowing these helps identify main features of the graph.
• Variability: Checking how the function increases or decreases across its domain can determine where it has maximum or minimum points.

Function analysis lays the groundwork for finding inverse points which are vital in creating a complete picture of the function and its inverse.

Inverse Points

Finding inverse points involves understanding the fundamental definition of inverse functions. For a function, if \((a, b)\) lies on its graph, then \((b, a)\) will be on the graph of its inverse. This concept flips the roles of inputs and outputs.

• Solving Equations: To locate inverse points for specific \(y\)-values (such as 0, 1, and 2 in our case), substitute these into the function and solve for \(x\). The values of \(x\) you find adjust to become \(y\) values on the inverse graph.
• Inverse Graph Points: Thus, solving \(f(x) = 0\), \(f(x) = 1\), and \(f(x) = 2\) gives us crucial points \((0, x_1)\), \((1, x_2)\), and \((2, x_3)\) marking the inverse.

Understanding inverse points bridges the connection between a function and its inverse, highlighting the relationship in their graphs.