Problem 24 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The function \(f(x)=2 x^{2}+x^{4}+1\) is even.

Step 1: Check if the function is even

To determine if this function is even, replace \(x\) with \(-x\) in the function:\(f(-x) = 2(-x)^2 + (-x)^4 + 1 = 2x^2 + x^4 + 1\)which is equal to the original function \(f(x)\). Therefore, \(f(x)\) is even.

Step 2: Check if the function is odd

As the function has already been found to be even, it cannot be odd. By definition, a function cannot be both even and odd at the same time.

Step 3: Formulate the conclusion

Since \(f(-x) = f(x)\), the function is even. There is no need to check if the function is odd, because it can’t be both even and odd. Therefore, the function \(f(x)=2 x^{2}+x^{4}+1\) is an even function.

Key Concepts

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to whole number powers and coefficients. These functions can have one or more terms, such as linear (degree 1), quadratic (degree 2), cubic (degree 3), and so on. The function given in the exercise is a polynomial of degree 4. This is because the highest power of the variable, which is '4', determines the degree of the polynomial.

These functions are significant due to their smooth and continuous nature, which makes them easy to differentiate and integrate. Understanding polynomial functions is crucial as they appear in different forms and applications in algebra and calculus. They serve as the foundation for various mathematical concepts and models in real-world situations.

• Terms - Individual parts of the polynomial expression, e.g., \(2x^2\), \(x^4\), and \(1\).
• Coefficients - Numerical values multiplying the variable terms, e.g., 1, 2.
• Degree - The highest power of the variable, indicating the polynomial's complexity and shape.

Functional Symmetries

Functional symmetries refer to whether a function is even, odd, or neither. Understanding these symmetries is essential as they provide insights into the graph's behavior around the y-axis and origin.

An even function is symmetric about the y-axis. To determine if a function is even, you replace \(x\) with \(-x\) in the function. If the result is identical to the original function, it is even, as we saw in the exercise with \(f(x) = 2x^2 + x^4 + 1\).

• Even function: \(f(-x) = f(x)\).
• Odd function: Symmetry about the origin, where \(f(-x) = -f(x)\).

Since a function cannot be both even and odd, this distinction helps decide which symmetrical properties the function graph should follow. If neither condition satisfies, the function classifies as neither even nor odd.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators, forming the backbone of algebra. In the context of functions, these expressions define the output corresponding to each input, such as in polynomial functions.

The given function \(f(x) = 2x^2 + x^4 + 1\) is an algebraic expression. It combines multiple terms, each consisting of a constant multiplied by the variable raised to a power. While working with algebraic expressions, it's crucial to understand how to manipulate them, substitute values, and more.

• Understanding terms - Recognizing each part of an algebraic expression aids in correctly evaluating and simplifying it.
• Substitution - Replacing variables with values or expressions is commonly used to evaluate expressions, as shown in the step-by-step solution where \(-x\) was substituted to check symmetry.

These expressions are not only essential in mathematical theory but in solving practical problems across various scientific fields.