Problem 65 - 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. \(f(x)=x^{3}\)

Step-by-Step Solution

Answer

The inverse of \( f(x) = x^3 \) is \( f^{-1}(x) = x^{1/3} \). Graph both functions and line of symmetry \( y = x \).

Step 1: Understand the Function

The given function is \( f(x) = x^3 \). This is a cubic function, which is one-to-one and thus has an inverse.

Step 2: Replace f(x) with y

Write the function as an equation by replacing \( f(x) \) with \( y \): \[ y = x^3 \]

Step 3: Swap x and y

To find the inverse, swap \( x \) and \( y \) in the equation. This gives: \[ x = y^3 \]

Step 4: Solve for y

Solve the equation \( x = y^3 \) for \( y \). Take the cube root of both sides: \[ y = x^{1/3} \]

Step 5: Write the Inverse Function

The inverse function is: \[ f^{-1}(x) = x^{1/3} \]

Step 6: Graph the Functions

Plot both \( f(x) = x^3 \) and \( f^{-1}(x) = x^{1/3} \) on the same coordinate system. Use a graphing tool or sketch manually.

Step 7: Draw the Line of Symmetry

The line of symmetry for a function and its inverse is \( y = x \). Draw this line on the same graph to show the symmetry between the function and its inverse.

Key Concepts

Understanding Cubic Functions

A cubic function is a type of polynomial function that has the general form \( f(x) = ax^3 + bx^2 + cx + d \). In this case, our given function is \( f(x) = x^3 \), which is a special type of cubic function without any quadratic, linear, or constant terms. This function is a simple cube function and is one of the most basic types.

Cubic functions have various characteristics:

• They can have either one or three real roots, based on their discriminant.
• They tend to have one inflection point where the concavity of the function changes.
• Their graphs are shaped like an elongated 'S' curve or a 'snake-like' path, reflecting the cube nature of the function.

Cubic functions typically increase as you move away from the origin, both in the positive and negative directions of the coordinate plane. The function \( f(x) = x^3 \) is symmetric with respect to the origin, exemplifying its odd nature, meaning \( f(-x) = -f(x) \).

Essentials of Function Graphing

Graphing functions like a pro involves understanding the behavior and layout of the function on the coordinate plane. When we graph the given cubic function \( f(x) = x^3 \), you'll notice it passing through points like \( (-2, -8) \), \( (0, 0) \), and \( (2, 8) \), showing its steady increase as \( x \) moves away from zero.

The inverse of \( f(x) = x^3 \) is \( f^{-1}(x) = x^{1/3} \), which has its own distinctive graphic properties. This function is the inverse operation, indicating a reflection over the line of symmetry \( y = x \). Graphing both functions together on the same set of axes reveals a perfect symmetry; you'll see one function literally flipping around the diagonal line \( y = x \).

To effectively chart these relationships:

• Plot sufficient points for each function to accurately capture their curves.
• Use a clean grid to facilitate precise drawing.

Having both functions visible on one system is an excellent way to visually understand the relationship between a function and its inverse.

The Role of Line of Symmetry

The line of symmetry, \( y = x \), plays a crucial role in understanding function inverses. It acts as the reflection point where any point \( (a, b) \) on the function \( f(x) \) will reflect to the point \( (b, a) \) on the inverse \( f^{-1}(x) \).

This symmetry line is particularly easy to draw — simply create a straight line with a 45-degree angle passing through the origin, ensuring that the coordinates are equal. The line helps to verify the correctness of the graphed inverses, as the inverse should ideally mirror across this line.

In practice:

• Ensure the line is accurately drawn by using the same scale on both x and y axes.
• Check that each function point and its inverse lie equidistant from the line of symmetry.

Understanding this relationship not only aids graphical accuracy but reinforces the fundamental concept of inverses and symmetrical properties in mathematical functions.