Problem 22 - Intermediate Algebra

Question

Determine whether each function is one-to-one. $$ f(x)=x^{3}+1 $$

Step-by-Step Solution

Answer

The function \( f(x) = x^3 + 1 \) is one-to-one as it passes the test \( f(a) = f(b) \Rightarrow a = b \) and is strictly increasing.

Step 1: Understand the Definition of One-to-One Function

A function is one-to-one if every output is determined by exactly one input. This implies that if \( f(a) = f(b) \), then \( a = b \). In other words, no two different inputs should result in the same output for the function.

Step 2: Analyze the Function's Formula

Consider the function \( f(x) = x^3 + 1 \). We need to determine if this function satisfies the condition \( f(a) = f(b) \) implies \( a = b \).

Step 3: Set Up the One-to-One Condition

Assume \( f(a) = f(b) \), which means \( a^3 + 1 = b^3 + 1 \). Simplifying this gives \( a^3 = b^3 \).

Step 4: Solve for a and b

From \( a^3 = b^3 \), take the cube root of both sides to get \( a = b \). This shows that if \( f(a) = f(b) \), then \( a = b \), which satisfies the condition for the function to be one-to-one.

Step 5: Verify by Considering the Function's Behavior

The function \( f(x) = x^3 + 1 \) is a cubic polynomial, which is known to be strictly increasing. For a strictly increasing (or decreasing) function, inputs relating to different outputs confirm that the function is one-to-one.

Key Concepts

Understanding Cubic Polynomials

A cubic polynomial is a type of polynomial function that can be described by an equation of the form \( f(x) = ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants, and the coefficient \( a \) is non-zero.
A common characteristic of cubic polynomials is that they can have up to three distinct real roots and always have at least one real root. This is due to the fact that it is a third-degree polynomial. Cubic polynomials can exhibit unique symmetrical properties and can be either increasing or decreasing based on their coefficients.

• An example is the cubic polynomial \( f(x) = x^3 + 1 \), which has no quadratic or linear term, and its constant is 1.
• If you expand this to see its full potential behavior, it primarily increases or decreases depending only on the \( x^3 \) term since other terms are absent or negligible in higher \( |x| \) values.

Knowing the general characteristics of cubic polynomials helps us predict their behavior and understand how they behave over the entire set of real numbers.

Exploring Function Behavior

The behavior of a function is a crucial component in understanding whether it's one-to-one. A function is considered one-to-one when it steadily increases or decreases without any plateaus or reversals. For \( f(x) = x^3 + 1 \), analyzing its behavior involves considering the derivative.

• Start by differentiating the function: \( f'(x) = 3x^2 \).
• Since \( 3x^2 \) is always positive or zero for any real \( x \) (never negative), this tells us that the function continuously increases.
• This continuous increase confirms that no two different \( x \) values will yield the same \( f(x) \) value.

This concept of continuous growth without turning back ensures each input in the function is mapped to a unique output, which is a key trait for one-to-oneness. Understanding function behavior through derivatives provides a strong mathematical framework for such characteristics.

Mathematical Proofs and Their Role

Mathematical proofs serve as the cornerstone of validating conjectures within mathematics. In the case of determining if a function is one-to-one, a proof can solidify the understanding. We used a proof by contradiction: assuming \( f(a) = f(b) \) and showing how this leads directly to \( a = b \).

Proving that \( f(x) = x^3 + 1 \) is one-to-one involved a few simple steps:

• Assume \( f(a) = f(b) \), which translates to \( a^3 + 1 = b^3 + 1 \).
• This simplifies to \( a^3 = b^3 \).
• The cube root of both sides gives \( a = b \).

By this understandable series of logical steps, it's demonstrated that if two outputs are equal, the inputs must also be equal—proving the function is indeed one-to-one. Mathematical proofs such as these reassure us of the function's properties and provide a consistent methodology that can be applied to analyze other functions too.