Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-16: Determine whether the function \(f\) is one-to-one. $$ f(x)=\sqrt[3]{x} $$

Step-by-Step Solution

Answer

Yes, the function is one-to-one.

Step 1: Understand the Concept of a One-to-One Function

A function is one-to-one if different inputs produce different outputs. Mathematically, if \(f(x_1) = f(x_2)\) implies \(x_1 = x_2\), then the function is one-to-one.

Step 2: Analyze the Given Function

The given function is \(f(x) = \sqrt[3]{x}\). This is the cube root function, which is one of the standard even functions.

Step 3: Determine Injectivity through Algebraic Manipulation

Assume \(f(x_1) = f(x_2)\), which means \(\sqrt[3]{x_1} = \sqrt[3]{x_2}\). To eliminate the radicals, raise both sides to the power of 3, resulting in \(x_1 = x_2\). This confirms the function's injectivity.

Step 4: Conclude the Analysis

Since \(x_1 = x_2\) whenever \(f(x_1) = f(x_2)\), the function \(f(x) = \sqrt[3]{x}\) is indeed a one-to-one function according to its algebraic property.

Key Concepts

Function Injectivity

The concept of injectivity is crucial in understanding one-to-one functions. A function is said to be injective, or one-to-one, if every element of the function's range is mapped by a unique element from its domain. In simpler terms, no two different inputs produce the same output. For a function \(f(x)\) to be one-to-one:

• If \(f(x_1) = f(x_2)\), then it must be true that \(x_1 = x_2\).

This property ensures that each output is linked to one and only one input. Think of injectivity as a way to make sure there are no duplicates in the output for any different inputs used. An injective function stands out because it never compresses different inputs into the same result. This characteristic is essential in many mathematical and real-world applications, such as encoding unique data points without overlap.

Cube Root Function

The cube root function is represented as \(f(x) = \sqrt[3]{x}\). It is a straightforward yet powerful mathematical function. The cube root of a number \(x\) is a value that, when cubed, gives back \(x\). This operation is inherently different from a square root due to its ability to handle all real numbers, including negatives.In the cube root function:

• Both positive and negative inputs are valid, as negatives have real cube roots.
• The function is symmetric around the origin, meaning the behavior for negative inputs mirrors that for positive inputs.
• It passes through the point (0, 0), as \(\sqrt[3]{0} = 0\).

Unlike the square root function, which only works for non-negative \(x\), the cube root function provides a smoothly increasing or decreasing curve, without breaks or gaps. This graph and symmetry indicate its natural one-to-one characteristic since each output in the range matches precisely one input.

Algebraic Manipulation

Algebraic manipulation is often used to demonstrate the properties of functions, including injectivity. For instance, with the given cube root function \(f(x) = \sqrt[3]{x}\), we can show it is injective via algebraic steps:To test injectivity algebraically, assume:

• \(f(x_1) = f(x_2)\)
• This translates mathematically to \(\sqrt[3]{x_1} = \sqrt[3]{x_2}\).

By performing a valid operation such as cubing both sides, the cube roots cancel, yielding:

• \(x_1 = x_2\)

This algebraic manipulation confirms that the function \(f(x) = \sqrt[3]{x}\) maps distinct inputs to distinct outputs, fulfilling the requirement for injectivity. Algebra provides a powerful toolkit to examine and prove the nature of various mathematical functions with precision and clarity.