Problem 6 - Precalculus

Question

A function \(f\) is ____ when, for each \(x\) in the domain of \(f, f(-x)=-f(x).\)

Step-by-Step Solution

Answer

The function f is odd.

Step 1: Understand the definition

A function is known as an odd function when the given condition, f(-x) = -f(x), is satisfied for all x in its domain. This means that if you replace x with -x in the function, the result is the negative of the original function.

Step 2: Match the definition

Looking at the condition mentioned in the problem, we see that it matches the definition of an odd function exactly. Thus, the word to be filled in the blank is 'odd'.

Key Concepts

Function Properties

Functions have many unique properties that help us understand their behavior and characteristics more clearly. One of these properties is related to how the function behaves when certain variables are changed or transformed.
For instance, consider how a function reacts when you substitute a variable with its opposite. This specific behavior leads us to categorize functions into certain types, such as odd functions. An odd function, as defined earlier, satisfies the condition \( f(-x) = -f(x) \).
Other common properties you might encounter include whether a function is increasing or decreasing, continuous or discontinuous, and bounded or unbounded. Knowing these properties can help you sketch graphs, solve equations, and understand the long-term behavior of the function.
To better grasp function properties, it’s helpful to frequently visualize functions and practice identifying these characteristics in different examples.

Mathematical Functions

In the realm of mathematics, functions are fundamental building blocks used to describe various relationships between sets of numbers or objects. A mathematical function is essentially a rule that assigns to each element in a set of inputs exactly one element in a set of outputs.
This relationship is often expressed as \( f : X \to Y \), where every element \( x \) in \( X \) (the domain) corresponds to one and only one element \( f(x) \) in \( Y \) (the range or codomain).
Functions can take on many forms, from simple linear functions like \( f(x) = 2x + 3 \) to complex trigonometric functions such as \( f(x) = \sin(x) \). Each type of function has its own set of behaviors and rules. Learning about these different kinds of functions allows students to tackle a variety of mathematical problems more effectively.

Function Symmetry

Symmetry in functions is a topic that reveals a lot about the structure and behavior of a function. It describes how certain transformations affect the function's graph. One of the key types of symmetry is around the origin, which is characteristic of odd functions.
An odd function, as we've covered, shows symmetry about the origin, meaning if its graph is rotated 180 degrees around the origin, it looks the same. This symmetry is a direct result of the unique property \( f(-x) = -f(x) \).

• This type of symmetry helps in predicting the function's values without plotting every point.
• Other types include symmetry about the y-axis, seen in even functions where \( f(-x) = f(x) \).

Recognizing the symmetry in functions will not only aid in graphing but also in solving equations and understanding functional behavior in a broader sense.