Problem 47 - College Algebra

Question

a. Find an equation for \(f^{-1}(x)\). b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\). $$ f(x)=(x+2)^{3} $$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \sqrt[3]{x} - 2\). Both the function and its inverse have a domain and range of all real numbers, that is (\(-\infty , \infty\)).

Step 1: Find The Inverse Function

To find the inverse function \(f^{-1}(x)\), replace \(f(x)\) with \(y\). So, we have \(y = (x+2)^3\). Now, swap \(x\) and \(y\), yielding \(x = (y+2)^3\). To solve for \(y\), take the cube root from both sides, which gives us \(y = \sqrt[3]{x} - 2\). Therefore, the inverse function is \(f^{-1}(x) = \sqrt[3]{x} - 2\).

Step 2: Graph \(f\) and \(f^{-1}\)

Graph the function \(f(x) = (x+2)^3\) by choosing a variety of x-values and calculating the corresponding y-values. Graph the inverse function \(f^{-1}(x) = \sqrt[3]{x} - 2\) in the same way, but note that \(y = \sqrt[3]{x}\) is the y-coordinate for any x-coordinate in the range of the original function.

Step 3: Determining the Domain and Range

The domain of a function is the set of all possible x-values and its range is the set of all corresponding y-values. For the function \(f(x) = (x+2)^3\), the domain is all real numbers (\(-\infty , \infty\)) and its range is also all real numbers (\(-\infty , \infty\)). For the inverse function, \(f^{-1}(x) = \sqrt[3]{x} - 2\), its domain is also all real numbers (\(-\infty , \infty\)) and the range it covers is also all real numbers (\(-\infty , \infty\))

Key Concepts

Understanding Domain and Range

The **domain** and **range** of a function are essential aspects to comprehend when working with any mathematical expression. The domain refers to all possible input values the function can accept. Meanwhile, the range comprises all potential output values derived from those inputs.

For the function given in our exercise,

• **Domain of** \(f(x) = (x+2)^{3}\): All real numbers, denoted by \((-\infty, \infty)\). This means there are no restrictions on what x-value we can plug into the cube function.
• **Range of** \(f(x)\): Likewise, all real numbers \((-\infty, \infty)\). A cube function can produce any real output.
• For its inverse, \(f^{-1}(x) = \sqrt[3]{x} - 2\): The domain and range remain the same, \((-\infty, \infty)\), as cube roots can handle any real number and adding or subtracting numbers doesn’t impose any restrictions either.

Grasping the domain and range prepares you to know what values you can expect from both the function and its inverse.

Exploring Cube Functions

Cube functions, like the one in our exercise, \(f(x) = (x+2)^{3}\), have a unique quality of being continuous and smooth for all real numbers.

Key Characteristics of Cube Functions:

• These functions rapidly grow as x moves away from zero, either positively or negatively.
• Being odd functions, these exhibit symmetric properties about the origin, meaning if you reflect the function across both the x-axis and y-axis, it appears unchanged.
• Cube functions always have a derivative except at infinity, meaning they do not have sharp turns anywhere on their graphs.

In the function above, the cubic power quickly amplifies small changes in x, reflecting the nature of such functions where transformations shift rather profoundly with altering x-values.

Graphing Functions

Graphing functions such as cube functions and their inverses can provide an insightful visual understanding. This process involves:

• **Plotting the Original Function:** For \(f(x) = (x+2)^3\), creating a table of x-values and their corresponding y-values (like -4, -3, 0, 1, 2, etc.), then plotting these points to visualise the curve.
• **Graphing the Inverse Function:** Similarly, plot \(f^{-1}(x) = \sqrt[3]{x} - 2\) using a similar point calculation method.
• The symmetry: When graphing both \(f(x)\) and \(f^{-1}(x)\) on the same axis system, you'll notice symmetry about the line y = x. This is a hallmark of inverse functions where each function can reflect onto the other.

Graphs help reveal the relationship between \(f\) and \(f^{-1}\), demonstrating how transformations affect their behavior and alignment in different ways. This pictorial view assists not just in computation but in broader mathematical insights as well.