Problem 14 - Algebra and Trigonometry

Question

Determine whether the function is one-to-one. \(f(x)=x^{4}+5, \quad 0 \leq x \leq 2\)

Step-by-Step Solution

Answer

Yes, the function is one-to-one on [0, 2].

Step 1: Check monotonicity on the given domain

\(f(x) = x^4 + 5\) on \([0, 2]\). The derivative is \(f'(x) = 4x^3\).

On \([0, 2]\): \(f'(x) = 4x^3 \geq 0\), with equality only at \(x = 0\).

Step 2: Conclude

Since \(f'(x) > 0\) for all \(x \in (0, 2]\), the function is strictly increasing on \([0, 2]\). A strictly increasing function is one-to-one.

The function \(\textbf{is one-to-one}\) on \([0, 2]\).

Key Concepts

Function Analysis

Function analysis involves examining the behavior and characteristics of a function to understand its nature and properties. In our exercise, we evaluate whether the function \(f(x) = x^4 + 5\) is one-to-one within the specified domain \(0 \leq x \leq 2\). To analyze a function for one-to-one properties, we consider if each value within its domain maps to a unique output value. In the case of this function, you would observe the graph or utilize the derivative to test for monotonicity.

• Graphical View: Examine if the graph of \(f(x) = x^4 + 5\) increases or decreases consistently within the domain, as a consistently increasing or decreasing function is one-to-one.
• Derivative Approach: The derivative, \(f'(x) = 4x^3\), can be analyzed to determine if it is non-negative or non-positive over the domain. The derivative \(f'(x) = 4x^3\) suggests that the function is consistently increasing for all \(x > 0\), confirming it is one-to-one on the given interval.

By understanding the behavior of the function through analysis, we can verify if it meets the requirement for being one-to-one.

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function \(f(x) = x^4 + 5\), the domain is given as \(0 \leq x \leq 2\). This means you focus only on the x-values ranging from 0 to 2, inclusive.

• Importance of Domain: The domain restricts where you look for inputs and outputs, as ensuring the function's defined behavior within these values is crucial for trustworthy analysis.
• Impact on Function Analysis: When verifying if a function is one-to-one, especially involving polynomial equations like \(x^4 + 5\), the defined domain guides your analysis. This domain specifies where to check for increasing or decreasing behavior.

Understanding the domain helps focus the analysis, whether working algebraically or graphically, ensuring conclusions about properties like one-to-oneness are accurately drawn.

Function Properties

Function properties are characteristics or attributes that define how a function behaves. For checking if \(f(x) = x^4 + 5\) is one-to-one, understanding the properties of polynomial functions such as monotonicity and evenness can be crucial.

• Even vs. Odd Functions: While even functions, like \(x^4\), typically have symmetric outputs across the y-axis, within a restricted domain \([0, 2]\), symmetry doesn't override a one-to-one test, implying diverse properties take precedence.
• Monotonicity: The property of being strictly increasing (monotonically increasing) or decreasing ensures no repeats in output, crucial for one-to-one verification. As previously determined from the derivative \(f'(x) = 4x^3\), the function rises consistently across the domain.

Knowledge of these properties assists in effective evaluation of the function, ensuring thorough understanding and correct derivation of one-to-oneness across specified domains.