Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Decide whether each function is one-to-one. $$y=2 x^{3}+1$$

Step-by-Step Solution

Answer

The function is one-to-one.

Step 1: Define One-to-One Function

A function is one-to-one if each input has a unique output, and no two different inputs have the same output. Mathematically, for a function \( f(x) \), \( f(a) = f(b) \) implies \( a = b \).

Step 2: Set Up the Function Equation

We are given the function \( y = 2x^3 + 1 \). To determine if the function is one-to-one, assume \( f(a) = f(b) \), then set \( 2a^3 + 1 = 2b^3 + 1 \).

Step 3: Simplify the Equation

Starting from \( 2a^3 + 1 = 2b^3 + 1 \), subtract 1 from both sides to get \( 2a^3 = 2b^3 \). Then, divide both sides by 2 to obtain \( a^3 = b^3 \).

Step 4: Solve for the Variable

Take the cube root of both sides of the equation \( a^3 = b^3 \), which gives \( a = b \). This implies that the initial assumption \( f(a) = f(b) \) leads to \( a = b \). Thus, no two different inputs produce the same output.

Step 5: Conclude One-to-One Property

Since we have proven that assuming \( f(a) = f(b) \) leads to \( a = b \) for all real numbers \( a \) and \( b \), the function \( y = 2x^3 + 1 \) is one-to-one.

Key Concepts

Unique Output

When dealing with one-to-one functions, it is important to understand the concept of unique outputs. A one-to-one function ensures that each input, often referred to as "a distinct input," corresponds to, at most, one specific output. This means that no two different inputs will yield the same output value.

For example, consider the function provided: \[ y = 2x^3 + 1. \]In this scenario, if the output is unique, then assuming \( f(a) = f(b) \) must lead us to \( a = b \). Hence, inputs \( a \) and \( b \) are not just corresponding to any random output. Each input has its own unique outcome, thus maintaining the one-to-one property.

This fundamental property is what differentiates one-to-one functions from other types of functions where inputs might repeat, leading to common outputs among different inputs. By ensuring that every input has a distinctive effect on the output, it guarantees a straightforward mapping from inputs to outputs.

Function Properties

The core properties of functions help determine their behavior and characteristics. For one-to-one functions, one key property is injectivity, which essentially confirms this direct input-to-output relationship without repetition or overlap of outputs for different inputs.

A function like \( y = 2x^3 + 1 \) must display such a property. To validate, we check if the function maintains that no two different values map to the same result. For this case, setting the outcomes \( 2a^3 + 1 = 2b^3 + 1 \) leads us to \( a = b \) as determined in the solution steps.

Additionally, one-to-one functions often are either consistently increasing or decreasing. This trait ensures continuous progression, reinforcing their defining property of unique mapping between input and output. Such functions do not change directions, so the above function is increasing as \( x \) increases, confirming its one-to-one nature.

Input-Output Relationship

Understanding the input-output relationship in functions is crucial for analyzing their characteristics and behavior. In a one-to-one function, this relationship is unique and non-repetitive.

For \( y = 2x^3 + 1 \), the connection between each input \( x \) and its resulting output \( y \) establishes this function as unique. Simplifying the equation \( 2a^3 + 1 = 2b^3 + 1 \) to \( a^3 = b^3 \) and resolving to find \( a = b \), you verify that each distinct \( x \) precisely correlates with one \( y \).

In essence, the pattern and structure of such a function ensure that every x-value you input results in a different y-value compared to other inputs. This principle is central not just in math, but in understanding any system where inputs result in outputs. It's the foundation for analyzing how changes in one area (input) bring direct results (output), helping predict outcomes accurately.