Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each function is one-to-one. $$f(x)=\sqrt[3]{x}$$

Step-by-Step Solution

Answer

The function \( f(x)=\sqrt[3]{x} \) is one-to-one.

Step 1: Understanding One-to-One Functions

A function is considered one-to-one if every output value is paired with exactly one input value. This means that for any two different input values, their output values must also be different.

Step 2: Checking the Definition for Given Function

Given the function is \( f(x)=\sqrt[3]{x} \), we need to test whether different inputs produce different outputs. If \( f(a) = f(b) \), then \( a = b \) must be true for \( f(x) \) to be one-to-one.

Step 3: Applying the One-to-One Test

Suppose \( f(a) = f(b) \), then \( \sqrt[3]{a} = \sqrt[3]{b} \). Since both sides are cube roots, the equation simplifies to \( a = b \). This confirms that different inputs will result in different outputs.

Step 4: Conclusion

Since \( a = b \) whenever \( f(a) = f(b) \), the function \( f(x) = \sqrt[3]{x} \) is indeed one-to-one.

Key Concepts

Function Properties

A function represents a specific relationship between a set of inputs and a set of outputs. One key property of functions is the "one-to-one" characteristic. A function is one-to-one, or injective, if every output is associated with exactly one distinct input. In simple terms:

• No two different inputs should yield the same output.
• For any numbers, if \( f(a) = f(b) \), then \( a = b \).

Understanding this property is crucial as it determines how functions behave in various mathematical contexts. For example, one-to-one functions have unique inverses because each output value ties back uniquely to a single input. This distinctiveness makes studying these functions particularly interesting and applicable to various fields.

Cube Root Function

The cube root function, represented as \( f(x) = \sqrt[3]{x} \), is an intriguing type of function. It is a type of root function, which involves finding a value that, when cubed, gives the number under the radical symbol. A few features of the cube root function include:

• Unlike square root functions, cube root functions are defined for all real numbers, including negative numbers.
• The graph of \( f(x) = \sqrt[3]{x} \) is symmetrical about the origin, showcasing a property called "odd symmetry."
• The cube root function is continuous and smoothly curves through all its domain.

These characteristics make the cube root function suitable for solving real-world problems where negative values can occur, such as calculating the volume or mass using measurements with negative values.

Injective Function

An injective function is another term for a one-to-one function. These functions are special because they maintain a distinct relationship between each input and output. When testing whether a function is injective:

• We verify that if \( f(a) = f(b) \), then it logically follows that \( a = b \).
• If this holds true across the function's entire domain, then the function is injective.

The given cube root function \( f(x) = \sqrt[3]{x} \) is injective, since each output \( y \) has a unique input \( x \). Thus, it passes the test for injectivity.Injective functions are helpful for finding unique solutions in problems, ensuring that each input corresponds uniquely to only one output without overlap. This property is crucial in fields like cryptography and information theory, where maintaining unique mappings is essential.