Problem 46 - 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. $$f(x)=x^{3}-x-2, y=1,2,3$$

Step-by-Step Solution

Answer

Points on the inverse are approximately (1, 1.38), (2, 1.81), and (3, 2.11).

Step 1: Understand the Function

The function given is \( f(x) = x^3 - x - 2 \). This is a cubic function that we will need to graph and analyze to find its inverse.

Step 2: Graph the Given Function

Using a graphing calculator, plot the function \( f(x) = x^3 - x - 2 \). Look for its shape and any important features like intercepts and symmetry as this will aid in understanding the inverse.

Step 3: Identify the Function's Inverse Conceptually

The inverse function of \( f(x) \) is found by swapping the x- and y-values. Therefore, if the function passes through a point \((a, b)\), the inverse will pass through \((b, a)\).

Step 4: Determine Points on the Inverse Function

Using the given \(y\)-coordinates for the inverse graph (\(y = 1, 2, 3\)), identify corresponding \(x\)-values on the graph of the original function. This will form points of the type \((x, y)\) on the graph \(f(x)\), but \((y, x)\) for \(f^{-1}(x)\).

Step 5: Calculate Points on the Inverse

Find points on the original graph where the x-coordinates correspond to the \(y\)-coordinates given for the inverse function:- For \(y = 1\), find \(x\) where \(f(x) = 1\).- For \(y = 2\), find \(x\) where \(f(x) = 2\).- For \(y = 3\), find \(x\) where \(f(x) = 3\). Calculate these using the calculator or solving numerically.

Step 6: Points Calculation

Using a calculator or graph, estimate:- \(f(x) = 1\) gives \(x \approx 1.38\)- \(f(x) = 2\) gives \(x \approx 1.81\)- \(f(x) = 3\) gives \(x \approx 2.11\)Thus, the corresponding points on the inverse are approximated as:- (1, 1.38)- (2, 1.81)- (3, 2.11)

Step 7: State the Final Points

The approximate points on the graph of the inverse function with \(y\)-coordinates 1, 2, and 3 are:- (1, 1.38)- (2, 1.81)- (3, 2.11)

Key Concepts

Graphing Functions

Graphing a function like \( f(x) = x^3 - x - 2 \) helps you visualize its shape and important features. Graphs depict how the values of \( y \) change with \( x \). This function is a cubic, which means it tends to have an "S"-shaped curve. When graphing functions, there are some key elements to look for:

• Intercepts: Points where the function crosses the x-axis and y-axis.
• Turning points: Places where the graph changes direction.
• Symmetry: Checking if the graph is symmetric can reveal certain properties about the function.

When graphing, changes in the shape like bends or stretches indicate important behaviors of the function. By plotting it using steps or a graphing tool, you can better predict its behavior. Remember, understanding the graph is crucial for finding an inverse, as it shows the relationship between \( x \) and \( y \).

Cubic Functions

Cubic functions are polynomials of degree three, looking something like \( f(x) = ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants. They are known for their distinctive "S" or "snake" shape as they can curve up and down due to having potentially several turning points. Key features of cubic functions include:

• Three roots: Real or complex solutions where the graph intersects the x-axis.
• Inflection point: A point where the curvature changes sign, giving the graph a symmetry.

Since cubic functions can increase and decrease differently in various sections, analyzing their graphs can be fascinating. The challenge with cubic functions is in finding exact roots, which sometimes requires numerical methods. With these functions, solutions to equations may require complex numbers, or may have simple linear factors. Knowing these properties will aid in understanding the inverse function's plot.

Using a Graphing Calculator

Graphing calculators are powerful tools that can aid in not only graphing functions but also in exploring their properties more thoroughly. They allow for detailed plotting, enabling users to see and analyze functions like \( f(x) = x^3 - x - 2 \) more accurately than manual graphing. When using a graphing calculator, you can:

• Check intercepts: Easily identify where the graph crosses the axes.
• Zoom and pan: Adjust views to closely examine details like turning points and behavior near the axes.
• Find inverses: Identify corresponding points on the inverse graph by checking swapped x- and y-values.

These calculators can significantly enhance understanding by also providing numerical solutions when appropriate, such as finding precise values for the inverse. With functions involving cubics, graphing calculators allow you to see the full scope of the curve, offering insight into both the broader and more detailed features of the function.