Problem 30 - College Algebra and Calculus: An Applied Approach

Question

Decide whether the function is even, odd, or neither. \(h(x)=x^{3}+3\)

Step-by-Step Solution

Answer

The function \(h(x) = x^{3} + 3\) is neither even nor odd.

Step 1: Define the function

Firstly, write down the function that you are given, which is \(h(x) = x^{3} + 3\).

Step 2: Evaluate h(-x)

Now, substitute \(-x\) for \(x\) in the function to get \(h(-x)\). Function then becomes \(h(-x) = (-x)^{3} + 3 = -x^{3} + 3\).

Step 3: Compare with h(x) and -h(x)

Then, compare \(h(-x)\) with \(h(x)\) and \(-h(x)\). The function \(h(-x) = -x^{3} + 3\) is not the same as \(h(x) = x^{3} + 3\), therefore the function is not even. Also, \(h(-x) = -x^{3} + 3\) is not equal to \(-h(x) = -(x^{3} + 3) = -x^{3} - 3\), therefore the function is not odd.

Step 4: Conclude the result

Since \(h(-x)\) does not match either \(h(x)\) or \(-h(x)\), we can deduce that the function \(h(x) = x^{3} + 3\) is neither even nor odd.

Key Concepts

Function Symmetry

When examining the nature of a function, understanding its symmetry is crucial in determining whether it's even, odd, or neither. Symmetry simplifies visualizing mathematical behaviors, but first, let's break down what these terms mean.

• Even Functions: A function is even if it satisfies the condition \(f(-x) = f(x)\) for all values of \(x\). Graphically, even functions are symmetric about the vertical y-axis. Common examples include trigonometric functions like cosine.
• Odd Functions: A function is odd if \(f(-x) = -f(x)\) for all \(x\). In terms of symmetry, odd functions are symmetric about the origin, meaning a 180-degree rotation returns the graph to its original position. Sine and tangent are typical odd functions.
• Neither: Functions that do not satisfy either condition are classified as neither even nor odd, lacking any specific symmetry.

For example, the function \(h(x) = x^3 + 3\) doesn't fit either criterion because \(h(-x) = -x^3 + 3\) isn't equal to \(h(x)\) or \(-h(x)\), showing it's neither even nor odd.

Algebraic Functions

Algebraic functions are composed of basic algebraic operations: addition, subtraction, multiplication, division, or raising to a power. They are often presented as polynomials, radical functions, or quotients of polynomials.

Algebraic functions have some defining traits:

• Standard Form: Generally, they take the form of \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where the coefficients \(a_n, a_{n-1}, \ldots, a_0\) are constants.
• Simplicity: They involve polynomial expressions and rational expressions, making them more straightforward to work with compared to transcendental functions (like logarithmic or exponential).

In our example, \(h(x) = x^3 + 3\), we see a simple algebraic expression where a cubic term is combined with a constant. Understanding these functions helps lay the groundwork for exploring broader mathematical concepts.

Polynomial Functions

Polynomial functions, a vital subclass of algebraic functions, form the backbone of many algebraic applications. They include terms with whole number exponents. Let's explore what makes them special.

• Structure: A polynomial is expressed as \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), where \(a_n eq 0\) and \(n\) is a non-negative integer. Each term consists of a coefficient and a power of \(x\).
• Degree: Defined by the highest power of \(x\) in the function. For \(h(x) = x^3 + 3\), the degree is 3, making it a cubic polynomial. Higher degrees generally mean more complexity.
• Behavior: Polynomial functions are continuous and smooth, with no gaps or sharp corners, making them predictable and easy to analyze.

In essence, polynomial functions provide a foundation for understanding complex relationships through their straightforward structure, as seen in the given function \(h(x) = x^3 + 3\). This specific polynomial is straightforward, yet demonstrates the key features that define polynomial behavior.