Problem 52 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the function is odd, even, or neither. \(h(x)=2 x-x^{3}\)

Step-by-Step Solution

Answer

The function is odd.

Step 1: Understand the Definitions

A function is **even** if for all values of \(x\), \(f(-x) = f(x)\) holds true. A function is **odd** if for all \(x\), \(f(-x) = -f(x)\) holds. If neither condition is satisfied, the function is **neither even nor odd**.

Step 2: Evaluate \(h(-x)\)

Given \(h(x) = 2x - x^3\), calculate \(h(-x)\) by substituting \(-x\) into the function: \[ h(-x) = 2(-x) - (-x)^3 = -2x + x^3 \]

Step 3: Check for Evenness

Compare \(h(x)\) and \(h(-x)\). For even functions, \(h(-x)\) should be equal to \(h(x)\). We have: \[ h(x) = 2x - x^3 \] \[ h(-x) = -2x + x^3 \] Since \(h(-x) eq h(x)\), the function is not even.

Step 4: Check for Oddness

Check if \(h(-x) = -h(x)\) to see if the function is odd:\[ h(-x) = -2x + x^3 \] \[ -h(x) = -(2x - x^3) = -2x + x^3 \] Since \(h(-x) = -h(x)\), the function is odd.

Step 5: Conclusion

Based on the checks above, the function satisfies the condition for being odd, where \(h(-x) = -h(x)\).

Key Concepts

Polynomial Functions

Polynomial functions are a fundamental concept in mathematics, often encountered in algebra. These functions are expressions that involve variables raised to whole number powers, along with coefficients. In simple terms, they look like this: - A sum of terms consisting of a constant times a power of a variable.
- The general form of a polynomial of degree n is: \[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 \] where \( a_n, a_{n-1}, \dots, a_0 \) are coefficients, and n is a non-negative integer.Polynomials may appear complex, but they follow simple rules of operations. The degree of the polynomial is determined by the highest power of the variable present.
These functions are smooth and continuous, which leads to their nice properties in mathematical analysis and calculus. - A linear polynomial is of degree 1, for example, \( f(x) = 2x + 3 \).- A quadratic polynomial has degree 2, like \( g(x) = x^2 - 4x + 4 \). - A cubic polynomial, such as \( h(x) = 2x - x^3 \), has a degree of 3, presenting more complex symmetry properties.

Function Symmetry

Understanding the symmetry of functions is essential in simplifying and analyzing mathematical expressions. Symmetry broadly refers to how a function behaves when its input values are altered, specifically reflected about an axis.
The two most common types of symmetry in functions are:- **Even Symmetry**: A function \( f(x) \) is even if it satisfies the equation \( f(-x) = f(x) \). Graphically, this means it is symmetrical about the y-axis. Common examples include quadratic functions, like \( g(x) = x^2 \), which produce mirror images on either side of the axis.- **Odd Symmetry**: A function is odd if \( f(-x) = -f(x) \). This indicates symmetry about the origin, meaning the function looks the same upside down from the origin. An example is \( h(x) = 2x - x^3 \), which when evaluated for \( -x \), yields \( -h(x) \).Functions that do not satisfy either of these conditions might have no symmetry at all. Checking for symmetry can simplify function analysis and help in sketching graphs quickly.
Always begin assessing symmetry by substituting \( -x \) and comparing with the original function to identify its nature.

Algebraic Functions

Algebraic functions form a broad category within mathematics. They encompass any function that can be constructed using algebraic operations, involving polynomials, roots, and coefficients. These operations include:- Addition- Subtraction- Multiplication- DivisionAlgebraic functions are a staple in algebra and precalculus courses. They serve as a foundational block for more complex mathematical concepts, including calculus and differential equations. Each algebraic function is defined by an expression involving constants and variables. **Polynomial functions**, including \( h(x) = 2x - x^3 \), are important types of algebraic functions.
Other algebraic functions could involve roots such as square roots or cube roots; for example, \( f(x) = \sqrt{x} + x^3 \).In practical scenarios, algebraic functions model a wide range of phenomena from natural sciences to economics, helping predict trends or solve equations that describe real-world problems.- For any given algebraic function, we often investigate properties like domain, range, and symmetry.
These insights help in understanding the function's graph and behavior in various mathematical contexts.