Problem 43 - College Algebra

Question

For the following exercises, find the inverse function. Then, graph the function and its inverse. $$ f(x)=x^{3}-1 $$

Step-by-Step Solution

Answer

The inverse function is \( f^{-1}(x) = \sqrt[3]{x + 1} \).

Step 1: Understand the Function

The given function is \( f(x) = x^3 - 1 \). This is a cubic function, which is one-to-one, meaning it has an inverse.

Step 2: Express f(x) as y

Start by replacing \( f(x) \) with \( y \) to express the equation as \( y = x^3 - 1 \). This helps in finding the inverse function.

Step 3: Swap x and y

To find the inverse, swap \( x \) and \( y \) in the equation. This gives us \( x = y^3 - 1 \).

Step 4: Solve for y

Isolate \( y \) by adding 1 to both sides: \( x + 1 = y^3 \). To solve for \( y \), take the cube root of both sides: \( y = \sqrt[3]{x + 1} \).

Step 5: Write the Inverse Function

The inverse function of \( f(x) \) is \( f^{-1}(x) = \sqrt[3]{x + 1} \).

Step 6: Graph the Function and Its Inverse

Graph \( f(x) = x^3 - 1 \). The graph is a cubic curve shifted down by 1 unit. Then, graph its inverse \( f^{-1}(x) = \sqrt[3]{x + 1} \), which is a cubic root curve shifted to the left by 1 unit. Note that the two graphs are reflections of each other across the line \( y = x \).

Key Concepts

Cubic Function

A cubic function is a type of polynomial function where the highest degree of the variable is three. It generally takes the form \( f(x) = ax^3 + bx^2 + cx + d \), where \( a, b, c, \) and \( d \) are constants, and \( a eq 0 \). In our original exercise, the function is \( f(x) = x^3 - 1 \), which means:

• The cubic term is \( x^3 \).
• There's no quadratic or linear term (since their coefficients are 0).
• It is shifted down by 1 due to the \(-1\) at the end.

Cubic functions can take various shapes but typically have an "S" curve due to their nature of changing direction. These functions are continuous, and they span from negative to positive infinity as \( x \) also ranges from negative to positive infinity. An important characteristic of a cubic function is that it is not symmetrical about the origin or the y-axis, but instead, its symmetry can sometimes be found through other transformations or shifts, like in our case.

One-to-One Function

In mathematics, a one-to-one function is a function where each element of the range is mapped to a unique element in the domain. This ensures that no horizontal line intersects the graph of the function more than once.

• For a function to have an inverse, it must be one-to-one.
• We determine this by using the Horizontal Line Test, which is a visual way to see that no horizontal line cuts through more than one point of the graph.

The given cubic function \( f(x) = x^3 - 1 \) is one-to-one because it passes the Horizontal Line Test. It is strictly increasing, which means that as \( x \) increases, \( f(x) \) also consistently increases, never repeating values. This property directly leads to the existence of a unique inverse function.

Reflection Across y = x

Reflecting a function across the line \( y = x \) means swapping the roles of \( x \) and \( y \). This is crucial when determining the inverse of a function. For the original function \( f(x) = x^3 - 1 \), its inverse \( f^{-1}(x) = \sqrt[3]{x + 1} \) is found by interchanging \( x \) and \( y \) in its equation and solving for \( y \). When graphed, the original function and its inverse will "mirror" each other along the identity line \( y = x \).

• The line \( y = x \) acts as a symmetry line.
• The point \( (a, b) \) on the original graph corresponds to \( (b, a) \) on the inverse.

This reflective symmetry helps visually verify that two functions are indeed inverses of each other.

Graphing Inverse Functions

Graphing both a function and its inverse on the same set of axes offers a tangible way to understand their relationship. For the function \( f(x) = x^3 - 1 \), we plot it as a transformed cubic curve, with a characteristic shape that bends and twists through the plane.

• Its inverse, \( f^{-1}(x) = \sqrt[3]{x + 1} \), follows a cubic root path. It appears as a gentler curve, shifted horizontally.
• The graphs intersect on the line \( y = x \), given their reflection.

The graphical approach not only demonstrates the inverse relationship but also emphasizes the crucial aspect of one-to-one functions allowing inverses. When graphing, shading the area of interest or using different colors makes this comparison even more visually intuitive for learners, enhancing comprehension of these inverse dynamics.