Problem 3 - Advanced Engineering Mathematics

Question

Determine whether the function is even, odd, or neither. $$ f(x)=x^{2}+x $$

Step-by-Step Solution

Answer

The function is neither even nor odd.

Step 1: Understand the Definitions

A function is even if \[f(-x) = f(x)\] for every \( x \) in its domain. It is odd if \[f(-x) = -f(x)\] for every \( x \) in its domain. If neither condition is satisfied, the function is neither even nor odd.

Step 2: Calculate f(-x)

Replace \( x \) in the function with \( -x \):\[f(-x) = (-x)^2 + (-x) = x^2 - x\]

Step 3: Compare f(-x) and f(x)

Compare \( f(-x) \) with \( f(x) \). We have:\[f(x) = x^2 + x\]\[f(-x) = x^2 - x\]\( f(-x) \) is not equal to \( f(x) \), so the function is not even.

Step 4: Check for Odd Function

Compare \( f(-x) \) with \(-f(x)\):\[-f(x) = -(x^2 + x) = -x^2 - x\]\[f(-x) = x^2 - x\]\( f(-x) \) is not equal to \(-f(x)\), so the function is not odd.

Step 5: Conclude

Since \( f(x) \) is neither even nor odd as it does not satisfy either condition, we conclude:The function \( f(x) = x^2 + x \) is neither even nor odd.

Key Concepts

Function Characteristics

Functions, like individuals, have distinct characteristics that help us to understand their behavior and properties. One of the first things to look at when dealing with functions is their domain and range. The domain is the set of all possible input values (often 'x' values), while the range consists of the potential output values (often 'f(x)' or 'y' values). Understanding these can shape how we view the function overall.
Additionally, other important characteristics include:

• Continuity: Is the function uninterrupted or does it have breaks?
• Intercepts: Where does the function intersect the axes?
• Asymptotes: Are there any lines the function approaches but never touches?
• End Behavior: How does the function behave as it approaches extreme values?

For polynomial functions, like in our exercise, understanding these characteristics can help us predict their shape and behavior graphically. Each term in a polynomial adds its own twist to the function's characteristics.

Function Symmetry

Symmetry in functions isn’t just a topic about beauty and form; it's crucial for understanding the fundamental structure of a function. Functions can be symmetrical, which usually means they are either even or odd, and sometimes neither.
In the context of functions:

• An even function has symmetry about the y-axis. This means it mirrors itself perfectly across the y-axis. The mathematical condition for an even function is that for every input, \(f(x) = f(-x)\).
• An odd function has rotational symmetry about the origin. This type of symmetry means if you spin the graph 180 degrees around the origin, it looks the same. Odd functions meet the condition that \(f(-x) = -f(x)\).

Not every function must be even or odd. Some functions, like in the exercise we've discussed, do not satisfy the conditions for either symmetry and hence are categorized as neither. It’s all about identifying patterns and understanding their implications for the function’s graph.

Polynomial Functions

Polynomial functions are expressions that involve sums of powers of a variable. They are incredibly versatile and widespread in algebra because they can model a vast range of real-world phenomena. The basic structure comprises terms of the form \(ax^n\), where \(a\) is a coefficient and \(n\) is a non-negative integer called the degree of the term. The degree of the polynomial is the highest power of the variable present in the function.
Key insights about polynomial functions include:

• The degree determines the shape. The highest degree term influences the end behavior.
• Roots or zeroes of the polynomial are solutions where the function equals zero, giving vital points where the graph intersects the x-axis.
• Each term in a polynomial can influence its symmetry: if all exponents are even, the function might be even; if odd, the function might be odd.

In our exercise, the function \(f(x) = x^2 + x\) is a polynomial of degree 2, as determined by its highest power term \(x^2\). The combination of different powers leads typical polynomial behaviors, yet it does not demonstrate symmetry that would make it even or odd. Understanding these nuances offers valuable insights into graphing and solving such functions.