Problem 30 - Algebra A Combined Function

Question

Each of the following functions is one-to-one. Find the inverse of each function and graph the function and its inverse on the same set of axes. $$ f(x)=x^{3}-1 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Function

We are given the function \( f(x) = x^3 - 1 \). Our task is to find its inverse. A function that is one-to-one always has an inverse.

Step 2: Replace f(x) with y

Start by rewriting the function \( f(x) = x^3 - 1 \) as \( y = x^3 - 1 \). This way, it will be easier to solve for the inverse.

Step 3: Solve for x in terms of y

We want to express \( x \) in terms of \( y \). Rearrange the equation:\[ y = x^3 - 1 \]Add 1 to both sides:\[ y + 1 = x^3 \]Take the cube root of both sides to solve for \( x \):\[ x = \sqrt[3]{y + 1} \]

Step 4: Write the Inverse

To express the inverse function, swap \( x \) and \( y \) in the equation to get:\[ f^{-1}(x) = \sqrt[3]{x + 1} \]This is the inverse of the original function.

Step 5: Graph the Functions

To graph the original function \( f(x) = x^3 - 1 \), plot it using points and note it has a typical cubic shape shifted down by 1 unit. To graph the inverse \( f^{-1}(x) = \sqrt[3]{x + 1} \), note it is similar to a cube-root function, shifted left by 1 unit. Both graphs should reflect over the line \( y = x \).

Key Concepts

One-to-One Function

A function is considered one-to-one if every output is produced by exactly one input. In simpler terms, no two different input values will produce the same output value.
This feature is important because only one-to-one functions have inverses that are also functions.
To determine if a function is one-to-one, you can use either the Horizontal Line Test on the graph of the function or check if it is either strictly increasing or strictly decreasing over its entire domain.

• The Horizontal Line Test states that if no horizontal line intersects the graph of the function more than once, the function is one-to-one.
• The function is strictly increasing if a larger input always gives a larger output.
• The function is strictly decreasing if a larger input always gives a smaller output.

Understanding these properties helps in confirming that a given function, such as the cubic function provided, has an inverse.

Graphing Functions

Graphing functions involves plotting the points that satisfy the function and connecting them to visualize the behavior of the function.
For example, when graphing the function \( f(x)=x^3 - 1 \), you plot points

• Choose a few values for \( x \)
• Calculate the corresponding values of \( y \) to form the points (x, y)
• Plot these points on a coordinate plane
• Draw a smooth curve through the points

The inverse function \( f^{-1}(x) = \sqrt[3]{x + 1} \) is plotted similarly, reflecting the original function's graph across the line \( y = x \).
This reflection illustrates how the roles of inputs and outputs swap in inverses.
By graphing both the original and inverse functions on the same set of axes, you can see their symmetric relationship across the line \( y = x \). This visual representation is key to understanding the concept of inverse functions.

Cubic Function

A cubic function is a type of polynomial function represented as \( f(x) = ax^3 + bx^2 + cx + d \).
The power of three for the highest term gives the function its name and distinct characteristics.
For the given function \( f(x) = x^3 - 1 \):

• The coefficient \( a = 1 \), which means the graph will have a typical cubic shape.
• The graph of \( x^3 \) forms a curve going from the bottom left to the top right.
• The \(-1\) shifts the entire graph down by 1 unit.

Cubic functions are one-to-one and continuously increasing, which means they pass the Horizontal Line Test and have inverses that are functions.
These properties make cubic functions straightforward for solving and graphing in many mathematical problems.

Cube Root Function

A cube root function is the inverse of a cubic function. It is expressed in the form \( f(x) = \sqrt[3]{x} \).
In the inverse function \( f^{-1}(x) = \sqrt[3]{x + 1} \), the expression shows how the output value is obtained by reversing the cubing process and accounting for any shifts.

The cube root function has unique properties:

• It is defined for all real numbers, providing outputs for every input value.
• It is a smooth, continuous curve that passes through the origin.
• The function is symmetric in behavior to a cubic function across the line \( y = x \).
• Shifting the graph left by 1 unit accounts for the addition within the function \( (x + 1) \).

The smooth curve of the cube root showcases how the function "unwinds" the cubing process, returning the value that was originally cubed.
This relationship is what makes inverse functions so practical for reversal and problem-solving tasks in mathematics.