Question

If \(f\) is an odd function, determine whether \(g(x)=-2 f\left(-\frac{x}{3}\right)\) is even, odd, or neither.

Short answer

The function \( g(x) \) is neither even nor odd.

Step-by-step solution

Understand the Definitions

Recall that an odd function satisfies the property: \[ f(-x) = -f(x) \] and an even function satisfies: \[ f(-x) = f(x) \]

Substitute -x into g(x)

Calculate \( g(-x) \) by substituting \( -x \) into the function \( g(x) = -2 f\left(-\frac{x}{3}\right) \). This gives: \[ g(-x) = -2 f\left(-\frac{-x}{3}\right) = -2 f\left(\frac{x}{3}\right) \]

Use the Odd Property of f

Since \( f \) is given to be an odd function, we can use the property \( f\left(\frac{x}{3} \right) = -f\left(-\frac{x}{3} \right) \). Substituting this in, we get: \[ g(-x) = -2 f\left(\frac{x}{3}\right) = -2\left(- f\left(-\frac{x}{3}\right)\right) \]

Simplify the Expression

Simplify \(-2\left(- f\left(-\frac{x}{3}\right)\right)\): \[ g(-x) = 2 f\left(-\frac{x}{3}\right) \] Compare this with \( g(x) = -2 f\left(-\frac{x}{3}\right) \).

Conclude Whether g(x) is Even, Odd or Neither

Observe that \( g(-x) \) is not equal to \( g(x) \) and \( g(-x) \) is not equal to \( -g(x) \). Therefore, \( g(x) \) is neither even nor odd.

Key concepts

even function

In mathematics, an even function is a type of function where the value of the function remains unchanged when the input is inverted. Formally, a function \(f(x)\) is considered even if it satisfies the condition: \[ f(-x) = f(x) \] for all values of \(x\) within the function's domain.

Examples of even functions include \(f(x) = x^2\) and \(f(x) = \cos(x)\).
This property implies that the graph of an even function is symmetric with respect to the y-axis. For every point \((x, y)\) on the graph, there is a corresponding point \((-x, y)\).

Understanding whether a function is even is critical when analyzing its behavior, simplifying expressions, and solving equations. Being aware of this property can also help in determining the function's integrability over symmetric intervals.

function properties

Functions exhibit various properties that can help us understand their behavior and characteristics. Key properties of functions include:

• Parity (Even or Odd): As discussed, even functions satisfy \(f(-x) = f(x)\) and odd functions satisfy \(f(-x) = -f(x)\). Odd functions are symmetric about the origin.
• Periodicity: Some functions repeat their values over regular intervals. For example, \(sin(x)\) and \(cos(x)\) are periodic functions with a period of \(2\pi\).
• Continuity: A function is continuous if it has no breaks, jumps, or holes. Formally, \(f(x)\) is continuous at a point \(c\) if \(\lim_{x \to c} f(x) = f(c)\).
• Differentiability: If a function has a derivative at every point in its domain, it is said to be differentiable. Differentiable functions are necessarily continuous, but the converse is not always true.

Recognizing these properties can simplify the process of solving and analyzing functions, making it easier to predict their behavior and apply appropriate mathematical techniques.

function transformation

Function transformations involve altering a function in various ways to obtain a new function. Common transformations include:

• Translation: Shifting a function horizontally or vertically. For example, \(f(x-h)\) represents a horizontal shift by \(h\) units, and \(f(x) + k\) is a vertical shift by \(k\) units.
• Scaling: Stretching or compressing a function vertically or horizontally. A vertical scaling by a factor of \(a\) is represented as \(af(x)\), and a horizontal scaling by a factor of \(b\) is represented as \(f(bx)\).
• Reflection: Flipping a function over a line. Reflecting over the x-axis is represented by \(-f(x)\), and reflecting over the y-axis is represented by \(f(-x)\).
• Combinations: These transformations can be combined to form more complex transformations. For instance, \(af(bx + h) + k\) represents a combination of stretching, shifting, and translating.

Transforming functions is useful in various applications, including graphing, solving equations, and modeling real-world scenarios.

algebraic functions

Algebraic functions are functions that can be expressed using basic algebraic operations such as addition, subtraction, multiplication, division, and taking roots. These functions include:

• Polynomial Functions: Expressions involving terms raised to whole-number exponents, such as \(f(x) = 2x^3 - 4x^2 + 3x - 1\).
• Rational Functions: Ratios of two polynomials, such as \(f(x) = \frac{3x^2 - 2}{x - 1}\).
• Root Functions: Functions involving roots, such as \(f(x) = x^{1/2}\) or \(f(x) = \sqrt{x}\).

These functions are characterized by their simple structured forms and their relationship to roots and exponents. Understanding algebraic functions is crucial as they frequently appear in various areas of mathematics, including algebra, calculus, and analysis.

Recognizing the type of algebraic function one is dealing with can lead to more efficient problem-solving and a deeper understanding of mathematical patterns and relationships.