Problem 23 - College Algebra

Question

Determine whether each function is even, odd, or neither. $$f(x)=x^{2}-x^{4}+1$$

Step-by-Step Solution

Answer

The function \(f(x)\) is even.

Step 1: Substitute -x into the function and simplify

Substitute \(x\) with \(-x\) in the function \(f(x)\) to get \(f(-x)\). This gives us \(f(-x) = (-x)^{2} - (-x)^{4} + 1\). Simplify this to get \(f(-x) = x^{2} - x^{4} + 1\).

Step 2: Compare f(x) with f(-x)

Next, compare \(f(x)\) with \(f(-x)\) to find out whether the function is even, odd, or neither. In this case, \(f(x) = x^{2} - x^{4} + 1\) and \(f(-x) = x^{2} - x^{4} + 1\). Since they are exactly the same, the function is even.

Step 3: Verifying that the function is not odd

Although we've already determined the function is even, for completion, it's good to check whether it's odd as well. For a function to be odd, it must satisfy \(f(-x) = -f(x)\). If we substitute this into the equation, we'll get \(x^{2} - x^{4} + 1 = -(x^{2} - x^{4} + 1)\), which simplifies to \(x^{2} - x^{4} + 1 = -x^{2} + x^{4} - 1\). As these are not equal, we can conclude that the function is not odd.

Key Concepts

Function Symmetry

Understanding function symmetry can help you easily determine if a function is even, odd, or neither. This symmetry is all about the way a graph folds over an axis or with respect to the origin. Let's explore further:

• An even function is symmetric with respect to the y-axis. This means if you fold the graph along the y-axis, both sides will align perfectly. Mathematically, for a function \( f(x) \) to be even, it must satisfy \( f(-x) = f(x) \) for all values of \( x \).
• On the other hand, an odd function is symmetric with respect to the origin. When folded through the origin, the graph aligns with itself when one part is rotated 180 degrees. For these functions, \( f(-x) = -f(x) \).
• If a function does not conform to either of the above symmetries, it is considered neither even nor odd.

For example, the given polynomial function \( f(x) = x^2 - x^4 + 1 \) is even since \( f(x) = f(-x) \). The symmetry can simplify understanding and predicting the behavior of functions.

Polynomial Functions

Polynomial functions are algebraic expressions involving terms in the form of \( ax^n \), where \( a \) is a coefficient and \( n \) is a non-negative integer exponent. These functions are incredibly versatile and form a crucial part of algebra.

• A linear polynomial has a degree of 1, like \( f(x) = 2x + 3 \).
• A quadratic polynomial has a degree of 2, for example, \( f(x) = x^2 - 4x + 4 \).
• A cubic polynomial has a degree of 3, such as \( f(x) = x^3 - x^2 + x - 1 \).
• Higher-degree polynomials follow similarly, with quartic polynomials having a degree of 4, which includes the given example \( f(x) = x^2 - x^4 + 1 \).

Each term in a polynomial function contributes to its overall shape and defining features, such as zeros, and turning points. It's essential to understand how these individual parts work together.

Algebraic Expressions

Algebraic expressions are like the building blocks of algebra dealing with numbers, variables, and arithmetic operations. They are used to represent polynomial functions amongst other mathematical phenomena.

• At their simplest, algebraic expressions can be single terms, like \( 7x \) or \( -3 \).
• More complex expressions involve sums and differences, for example, \( x^2 + 5x - 9 \).
• Algebraic expressions can also involve multiple variables and exponents, which are combined using operations like addition, subtraction, multiplication, and division (except division by zero).

The key to mastering algebraic expressions is understanding how to manipulate and simplify them, which is a fundamental skill in solving equations, like identifying symmetries in functions. This understanding forms the backbone of work with polynomial functions and more advanced algebraic concepts.