Problem 66 - Elementary and Intermediate Algebra

Question

Find the inverse of each function. Then graph the function and its inverse on one coordinate system. Show the line of symmetry on the graph. \(f(x)=x^{3}+1\)

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt[3]{x - 1} \). Graph both functions and show symmetry around the line \( y = x \).

Step 1: Set Up Equation for Inverse Function

To find the inverse function, first write the function equation as: \( y = x^3 + 1 \). Then, interchange \( x \) and \( y \) to find the inverse: \( x = y^3 + 1 \).

Step 2: Solve for y

To solve for \( y \), isolate \( y \) on one side of the equation: subtract 1 from both sides to get \( x - 1 = y^3 \). Then take the cube root of both sides: \( y = \sqrt[3]{x - 1} \). Hence, the inverse function is \( f^{-1}(x) = \sqrt[3]{x - 1} \).

Step 3: Verify the Inverses

To verify, compose the function \( f(f^{-1}(x)) \) and see if it equals \( x \): \( f(\sqrt[3]{x-1}) = (\sqrt[3]{x-1})^3 + 1 = x - 1 + 1 = x \). Additionally, \( f^{-1}(f(x)) \) also results in \( y = x \), confirming that these are inverses.

Step 4: Graph the Function and Its Inverse

Plot \( f(x) = x^3 + 1 \) and \( f^{-1}(x) = \sqrt[3]{x - 1} \) on the same coordinate plane. Use key points to plot these functions: \((0, 1)\), \((1, 2)\), and \((-1, 0)\) for \( f(x) \), and their respective inverse points \((1, 0)\), \((2, 1)\), and \((0, -1)\) for \( f^{-1}(x) \).

Step 5: Draw Line of Symmetry

The line of symmetry for a function and its inverse is the line \( y = x \). Draw this line on the graph to verify that \( f(x) \) and \( f^{-1}(x) \) are symmetric about \( y = x \).

Key Concepts

Graphing Functions

Graphing functions is an essential step in visualizing mathematical relationships.
They help us see how one variable affects another in a given function. A function like \( f(x) = x^3 + 1 \) involves graphing a curve based on the equation provided.
Here are some key points to remember when graphing functions:

• Identify the nature of the function (linear, quadratic, cubic, etc.).
• Calculate key points by substituting several \( x \)-values into the equation and finding their corresponding \( y \)-values.
• Plot these points on a coordinate plane, then connect the points to get the entire graph.
• Understand the general shape and behavior of the function to sketch the curve accurately.

In this case, our function \( f(x) = x^3 + 1 \) is a cubic function, which we will discuss in more detail.
By graphing both \( f(x) \) and its inverse, \( f^{-1}(x) = \sqrt[3]{x - 1} \), you can visually confirm their relationship.

Cubic Functions

Cubic functions are polynomial functions where the highest degree is three. They can be written in the general form \( f(x) = ax^3 + bx^2 + cx + d \).
In the exercise, we're dealing with a simple cubic function: \( f(x) = x^3 + 1 \).Key characteristics of cubic functions include:

• The presence of an inflection point, where the curve changes concavity.
• They can have one or three real roots, depending on the discriminant.
• Typically, they produce an 'S' or reversed 'S' curve pattern on a graph.
• The end behavior will show each direction of the curve extending towards positive or negative infinity as \( x \) approaches infinity.

Cubic functions are non-linear and exhibit more complicated behavior compared to linear and quadratic functions.
In our cubic function \( x^3 + 1 \), the graph shifts upwards by 1 unit from the typical \( x^3 \) graph due to the constant \( +1 \).
Understanding these patterns helps us to draw the graph accurately and to determine its inverse function graph.

Line of Symmetry

In mathematics, a line of symmetry refers to a line that divides a figure or graph into two mirror-image sections.
For functions and their inverses, the line of symmetry plays a critical role. When you find the inverse of a function, such as switching \( x \) and \( y \) in the equation, the graphs of these functions are symmetrical along the line \( y = x \).
Important points about the line of symmetry in this context:

• It is an important tool to visually confirm if you've correctly plotted a function and its inverse.
• The line \( y = x \) means that at any point \( (x, y) \), swapping \( x \) and \( y \) will generate the corresponding point on the inverse function.
• Graphically, this is horizontal between \((0, 0)\) in all four quadrants and helps ensure both functions mirror each other accurately.

While graphing the function \( f(x) = x^3 + 1 \) and its inverse \( f^{-1}(x) = \sqrt[3]{x - 1} \), you should draw this line on the same coordinate plane.
This step visually validates that the function and its inverse are symmetric.