Problem 47 - Elementary and Intermediate Algebra

Question

Each of the following functions is one-to-one. Find the inverse of each function and express it using \(f^{-1}(x)\) notation. \(f(x)=\sqrt[3]{x}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to find the inverse of the function \( f(x) = \sqrt[3]{x} \). The inverse function, \( f^{-1}(x) \), will reverse the effect of \( f(x) \).

Step 2: Replace f(x) with y

Since \( f(x) = \sqrt[3]{x} \), we can write this as \( y = \sqrt[3]{x} \) to make it easier to find the inverse.

Step 3: Switch Variables x and y

To find the inverse, swap \( x \) and \( y \) in the equation. This gives us \( x = \sqrt[3]{y} \).

Step 4: Solve for y

To isolate \( y \), we need to undo the cube root. Cube both sides of the equation: \( x^3 = y \).

Step 5: Express the Inverse Function

Now that we have \( y = x^3 \), we can express this as the inverse function: \( f^{-1}(x) = x^3 \).

Key Concepts

One-to-One Function

In mathematics, a one-to-one function, also known as an injective function, is a type of function where each element of the domain maps to a unique element in the codomain. This means no two different elements in the domain are mapped to the same element in the range. Understanding whether a function is one-to-one is crucial when dealing with inverse functions, as only one-to-one functions have inverses that are also functions.

To identify a one-to-one function, you can use the horizontal line test. If any horizontal line drawn across the graph of the function only intersects it at one point, then the function is one-to-one:

• Each input (x-value) corresponds to one unique output (y-value).
• This property is essential for the function to have an inverse that is also a function.

For example, the function \(f(x) = \sqrt[3]{x}\) is one-to-one because for every unique \(x\), there's a unique \(\sqrt[3]{x}\), and thus it passes the horizontal line test perfectly.

Cube Root

The cube root of a number \(x\) is another number \(b\) such that when \(b\) is multiplied by itself twice (i.e., cubed), the result is \(x\). In simpler terms, if \(b^3 = x\), then \(b\) is the cube root of \(x\). The cube root function is particularly interesting because it can accept both positive and negative numbers, making it more versatile for solving various mathematical problems.

The cube root function is denoted as \(\sqrt[3]{x}\):

• A cube root function undoes the cubing process.
• The graph of \(\sqrt[3]{x}\) passes through the origin (0,0) and is symmetric with respect to the origin, illustrating that it is one-to-one.

Cubing and taking the cube root are inverse operations. Hence, when finding the inverse of a cube root function, like in the problem given, you end up switching \(x\) and \(y\) to solve for the inverse, ultimately leading to the expression \(f^{-1}(x) = x^3\).

Algebraic Functions

Algebraic functions are mathematical expressions constructed using algebraic operations such as addition, subtraction, multiplication, division, and taking roots. They form a broad class of functions that include familiar polynomial, rational, and root functions.

An important subtype of algebraic functions is those that involve roots, such as the square root or cube root, which are common in many mathematical contexts:

• These functions can often be described by equations involving polynomials.
• For example, the function \(f(x) = \sqrt[3]{x}\) is an algebraic function because it involves taking the cube root of \(x\).

Algebraic functions are fundamental in forming inverse functions. When dealing with inverses, understanding the operations involved allows you to "reverse" these operations, rewriting the function so that it swaps inputs and outputs. This principle is essential for solving equations where the function's structure needs to be preserved but inverted, allowing for deeper mathematical exploration and applications.