Problem 93 - Precalculus Student Solutions Manual 5th

Question

Use a graphing calculator to graph each function defined as follows, using the given viewing window. Use the graph to decide which functions are one-to-one. If a function is one-to-one, give the equation of its inverse. $$\begin{array}{l} f(x)=6 x^{3}+11 x^{2}-6 \\ {[-3,2] \text { by }[-10,10]} \end{array}$$

Step-by-Step Solution

Answer

Graph the function to check if it passes the horizontal line test for one-to-one determination. If one-to-one, find the inverse equation.

Step 1: Define the function

The function to be graphed is given by \[ f(x) = 6x^3 + 11x^2 - 6 \].

Step 2: Set the viewing window

The specified viewing window for the graph is \([-3, 2]\text{ by }[-10, 10]\). Ensure the graphing calculator is set to this window to correctly visualize the function.

Step 3: Graph the function using the calculator

Enter the function \(6x^3 + 11x^2 - 6\) into the graphing calculator and plot the graph using the viewing window \([-3, 2]\text{ by }[-10, 10]\). Observe the plotted graph carefully.

Step 4: Determine if the function is one-to-one

To determine if the function is one-to-one, check if the graph passes the horizontal line test. If any horizontal line intersects the graph at more than one point, then the function is not one-to-one.

Step 5: Conclusion about the function

By observing the graph in the specified window, determine if any horizontal line intersects the graph more than once. If such intersections exist, the function is not one-to-one. Otherwise, it is one-to-one.

Step 6: Find the inverse (if one-to-one)

If the function is determined to be one-to-one, find the equation of its inverse. This would involve solving the equation \( y = 6x^3 + 11x^2 - 6 \) for \( x \) in terms of \( y \), then switching \( x \) and \( y \) to find the inverse \( f^{-1}(x) \).

Key Concepts

graphing calculator

Graphing calculators can be super helpful tools in understanding the behavior of complex functions. When dealing with functions like \( f(x) = 6x^3 + 11x^2 - 6 \), it is essential to graph them correctly to observe their nature.
Start by inputting the function into the graphing calculator. Make sure the viewing window is set correctly, in this case, \([-3, 2]\text{ by }[-10, 10]\).
This range allows you to see the function’s behavior within those x and y intervals. You can adjust the window settings using your graphing calculator’s instructions to ensure you get an accurate and comprehensive view of the function. Visualizing the function graphically can make it easier to determine if it is one-to-one.

inverse functions

An inverse function essentially 'reverses' the effects of a function. If you have a function \(f(x)\), its inverse \(f^{-1}(x)\) undoes what \(f(x)\) does. Graphically, the inverse function reflects the original function across the line \(y = x\).
The main requirement for a function to have an inverse is that it must be one-to-one.
This means every x-value has a unique y-value, and every y-value has a unique x-value when looking at the function and its inverse.

horizontal line test

The horizontal line test is a simple yet powerful visual tool to determine if a function is one-to-one. To perform this test, draw horizontal lines across your graph of the function.
If any horizontal line intersects the graph at more than one point, then the function is not one-to-one. This would mean that the function does not have a unique inverse function.
For the function \( f(x) = 6x^3 + 11x^2 - 6 \), observe whether any horizontal line touches the graph more than once.
If no horizontal line does, then your function is one-to-one and you can proceed to find the inverse.

cubic functions

Cubic functions are functions of the form \( f(x) = ax^3 + bx^2 + cx + d \). They can exhibit complex behaviors such as turning points, inflection points, and varying rates of change.
In this exercise, \( f(x) = 6x^3 + 11x^2 - 6 \) is a cubic function. To analyze it, pay attention to these key features:

• Turning points: Places where the function changes direction.
• Inflection points: Where the curvature of the function changes.

These properties make cubic functions interesting and sometimes tricky to analyze, requiring tools like graphing calculators for accurate visualization.