Problem 35 - College Algebra with Corequisite Support

Question

For the following exercises, find the inverse of the function and graph both the function and its inverse. $$ f(x)=x^{3}+3 $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the function

The given function is \( f(x) = x^3 + 3 \). This function is a cubic function, which is one-to-one, so it should have an inverse function.

Step 2: Replace f(x) with y

Let's rewrite the function in terms of \( y \). \[ y = x^3 + 3 \]

Step 3: Solve for x

Our goal is to solve for \( x \) in terms of \( y \). This involves isolating \( x \) on one side of the equation.Subtract 3 from both sides:\[ y - 3 = x^3 \]Take the cube root of both sides:\[ x = \sqrt[3]{y - 3} \].

Step 4: Write the inverse function

Now, substitute \( y \) with \( x \) to write the inverse function. The inverse of \( f(x) \) is:\[ f^{-1}(x) = \sqrt[3]{x - 3} \].

Step 5: Graphing the functions

Graph \( y = x^3 + 3 \), which is a cubic graph shifted up by 3 units. Also graph \( y = \sqrt[3]{x - 3} \), which is the reflection of the original function across the line \( y = x \). This line will pass through any point where the original and inverse functions intersect.

Key Concepts

Cubic Functions

Cubic functions are a form of polynomial functions where the highest exponent of the variable is 3. The general format for a cubic function can be written as \( f(x) = ax^3 + bx^2 + cx + d \). In the given problem, the cubic function is simplified to \( f(x) = x^3 + 3 \), which means the graph of this function is a standard cubic curve moved upward by 3 units along the y-axis.

Cubic functions have the characteristic shape of an "S-curve," which means they have one inflection point where the curvature changes direction. The graph of a basic cubic function like \( x^3 \) is symmetrical around the origin, but changes in position can occur due to constant shifts. The function \( x^3 + 3 \), as in our case, maintains the S-curve shape but shifts vertically upward by 3 units.

• Unlike quadratic functions, cubic functions are odd, meaning they have rotational symmetry around the origin.
• They are continuous and smooth, with no breaks or sharp edges.
• The domain and range both are all real numbers, \((-\infty, +\infty)\).

Graphing Functions

Graphing functions visually represents the relationship between the input and output of a function. To graph a cubic function like \( f(x) = x^3 + 3 \), we start by plotting points. By selecting different \( x \) values and calculating corresponding \( y \) values, one can see the distinctive S-curve pattern of the cubic function. For the graph of \( y = x^3 + 3 \):

- Choose values of \( x \) around zero to see the vertical transform.
- Every resulting \( y \) value is shifted 3 units up from the standard cubic \( y = x^3 \) graph.
Graphing the inverse function also follows a systematic strategy. The inverse function \( f^{-1}(x) = \sqrt[3]{x - 3} \) reflects the original function across the line \( y = x \). This means:

• The graph of the inverse is obtained by swapping the role of \( x \) and \( y \).
• This creates a reflection that allows shared points, making it easy to visually confirm if the original and the inverse are indeed reflections.
• This shared line serves as a guide when plotting both functions simultaneously.

One-to-One Functions

A one-to-one function is a type of function in which every element of the range corresponds uniquely to exactly one element of the domain. This is a crucial property because only one-to-one functions have inverse functions that are also functions. The given cubic function, \( f(x) = x^3 + 3 \), is one-to-one.

To confirm a function is one-to-one, use the horizontal line test. In this test, if any horizontal line crosses the graph of the function at more than one point, the function is not one-to-one.
For cubic functions:

• Their graphs usually pass the horizontal line test. Due to their continuous and unrestricted growth in both positive and negative directions, they naturally satisfy this condition.
• This particular feature of cubic functions means they automatically have inverse functions that are well-defined.

The process of finding the inverse, as detailed in the solution, involves resolving for \( x \) in the original function equation. Since the original function is one-to-one, we directly find and verify its inverse function, ensuring it accurately maps inputs from the range back to the domain.