Problem 47 - College Algebra with Corequisite Support

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. $$ f(x)=x^{3}+x-2, y=0,1,2 $$

Step-by-Step Solution

Answer

The inverse function has points approximately at (0, root of f(x)=0), (1, root of f(x)=1), and (2, root of f(x)=2).

Step 1: Graph the Function

Use a calculator to plot the function \( f(x) = x^3 + x - 2 \). Observe the shape and behavior of the graph as it crosses the x-axis and where it turns.

Step 2: Understanding the Inverse

The inverse of a function \( f(x) \), denoted as \( f^{-1}(y) \), is found by switching the x and y coordinates. This means if \( (a, b) \) is on \( f(x) \), then \( (b, a) \) will be on \( f^{-1}(y) \).

Step 3: Identify Points with Given y-coordinates on f(x)

Look at the graph of \( f(x) \) and find the x-coordinates that correspond to the given y-coordinates 0, 1, and 2. This means solving \( f(x) = 0 \), \( f(x) = 1 \), and \( f(x) = 2 \) respectively.

Step 4: Solve f(x) = 0

Find the x-value where \( f(x) = 0 \). This roots can be found by analyzing or using the calculator's solve feature. Suppose it results in \( x = a \). So, \( (a, 0) \) is on \( f(x) \), therefore, \( (0, a) \) is on the inverse.

Step 5: Solve f(x) = 1

Determine the x-value by solving \( f(x) = 1 \). This x-value could be \( x = b \). The point \( (b, 1) \) on \( f(x) \) leads to \( (1, b) \) on \( f^{-1}(y) \).

Step 6: Solve f(x) = 2

Solve for the x-value when \( f(x) = 2 \). This might be \( x = c \). Thus, \( (c, 2) \) is on \( f(x) \) and \( (2, c) \) is a corresponding point on the inverse.

Key Concepts

Graphing Techniques

To master inverse functions, we begin by graphing the original function, in this case, \( f(x) = x^3 + x - 2 \). Graphing helps to visualize the function's behavior, such as turning points and the overall shape. When using a graphing calculator or plotting software, it is crucial to set an appropriate window that captures these characteristics.

When graphing:

• Observe where the graph intersects the x-axis. These are the real roots of the equation \( f(x) = 0 \).
• Note the turning points or where the graph changes direction; this often occurs when the function reaches a local minimum or maximum.
• Check for symmetry. While this specific function doesn’t have symmetric properties, some functions do, which can simplify finding an inverse.

These insights will help when transitioning from the graph to finding inverse values.

Function Analysis

Understanding the original function \( f(x) = x^3 + x - 2 \) is key to finding its inverse. Function analysis allows us to determine how the function behaves, which further guides us in identifying meaningful points for the inverse.

Key aspects of function analysis include:

• Continuous and Differentiability: The function is continuous and differentiable, essential properties for ensuring an inverse function exists in practical terms.
• Monotonicity: If the function is strictly increasing or decreasing, it guarantees the existence of an inverse since there are no repeating y-values for different x-values.
• Critical Points: Calculating where the first derivative equals zero helps find turning points, indicating where the function might change from increasing to decreasing or vice versa.
• End Behavior: Understanding how \( f(x) \) behaves as \( x \rightarrow \pm\infty \) helps in predicting the graph's trend, which is mirrored in the inverse function behavior.

These analytical aspects are vital in comprehending how changes in y-values correspond to specific x-values necessary for inverse function determination.

Root Finding

Finding the x-coordinates that match specific y-values greatly aids in determining points on the inverse of the function. With the given function \( f(x) = x^3 + x - 2 \), find the roots for \( f(x) = 0 \), \( f(x) = 1 \), and \( f(x) = 2 \).

Steps for root finding include:

• Using graphing techniques, identify initial estimates where the function touches or crosses y-values 0, 1, and 2.
• Numerical Methods: Employ methods like the bisection method, Newton's method, or use graphing calculators’ solve functions to accurately pinpoint roots.
• Verification: Once potential roots are found, plug them back into the function to verify that they result in the given corresponding y-values.

Successful root finding provides the required x-values; namely, \( (x_0, 0), (x_1, 1), (x_2, 2) \). Switching the x and y in these coordinates gives the points for the inverse function: \( (0, x_0), (1, x_1), (2, x_2) \). This completes the calculation for plotting the inverse. Understanding these techniques simplifies the process of moving from the given function to its inverse.