Problem 25 - Precalculus

Question

In Exercises \(21-41,\) determine analytically if the following functions are even, odd or neither. $$ f(x)=4-x^{2} $$

Step-by-Step Solution

Answer

The function is even.

Step 1: Definition of Even and Odd Functions

A function is called **even** if for every input \(x\), \(f(x) = f(-x)\). A function is called **odd** if for every input \(x\), \(f(-x) = -f(x)\). If a function satisfies neither condition, it is **neither even nor odd**.

Step 2: Analyze the Given Function

The given function is \( f(x) = 4 - x^2 \). We will test this function for evenness and oddness by substituting \(-x\) into the function.

Step 3: Substitute \(-x\) into the Function

Find \(f(-x)\) by substituting \(-x\) for \(x\) in the function: \[ f(-x) = 4 - (-x)^2 \]. Since \((-x)^2 = x^2\), this simplifies to \( f(-x) = 4 - x^2 \).

Step 4: Compare \(f(x)\) and \(f(-x)\)

Now we have \(f(x) = 4 - x^2\) and \(f(-x) = 4 - x^2\). Since \(f(x) = f(-x)\), the function satisfies the condition for being even.

Step 5: Conclusion

The function \(f(x) = 4 - x^2\) is even, as it satisfies the condition \(f(x) = f(-x)\) for all values of \(x\).

Key Concepts

Understanding Function Analysis

Function analysis is an essential area of mathematics that helps us observe and understand the behavior of functions in detail. It involves looking at various characteristics of mathematical functions to deduce their nature and complexities. Function analysis is crucial when determining whether a function is even, odd, or neither.
This kind of analysis requires us to substitute and compare values. By doing so, we can see how different inputs and their opposites affect the function.
For example, by substituting \(-x\) for \(x\) in the function, we can analyze the behavior of the function across the input domain. With careful observation and comparison of the results, we can identify patterns or symmetry that help characterize the function correctly.

Analytical Determination of Even and Odd Functions

Analytical determination refers to using algebraic methods to uncover properties of a function.
When analyzing functions to determine if they are even or odd, an algebraic approach involves substituting negative inputs and comparing results.

• For **even functions**, we check if the function satisfies the condition: \(f(x) = f(-x)\).
• For **odd functions**, we verify if the function holds: \(f(-x) = -f(x)\).
• If neither condition is satisfied, the function is classified as neither even nor odd.

In our example, the function \(f(x) = 4 - x^2\), substituting \(-x\) resulted in \(f(-x) = 4 - x^2\), which matches \(f(x)\). This analytical determination confirms that the function is even.
Such methodical approaches help ensure thorough understanding and accurate classification of mathematical functions.

Exploring Mathematical Functions

Mathematical functions are expressions that define relationships between numbers or variables. They serve as foundational tools in mathematics for modeling real-world phenomena and solving problems.
Functions can be linear, quadratic, polynomial, or more intricate forms, each distinctly impacting their roles and behaviors.
The distinction between even and odd functions specifically deals with symmetry:

• **Even functions** have a symmetric pattern around the y-axis. This symmetry means if you flip the graph horizontally, the function remains unchanged.
• **Odd functions** possess rotational symmetry around the origin. If you rotate the graph 180 degrees, it looks the same.
• Functions that do not fit either category neither exhibit these symmetries.

In understanding these types of functions, learners get insights into more comprehensive aspects of mathematical analysis, providing a strong framework to tackle problems efficiently.