Question

Determine whether the given function is periodic. If so, find its fundamental period. $$ x^{2} $$

Short answer

If yes, what is its fundamental period? Answer: No, the function \(f(x) = x^2\) is not a periodic function, as there is no non-zero value of T for which the function repeats itself.

Step-by-step solution

Understand the function

We have the quadratic function \(f(x) = x^2\). Recall that a periodic function repeats its values in regular intervals or periods.

Search for a Period

We will search for a non-zero value of T such that \(f(x + T) = f(x)\) for all x. Let's analyze the expression using the properties of exponents: $$ f(x + T) = (x + T)^2 \\ (x + T)^2 \stackrel{?}{=} x^2 \quad (\text{testing if equal for all } x) $$ For any arbitrary real number x, there is no fixed non-zero value \(T\) that satisfies the equation above for all x. Therefore, we can conclude that the function is not periodic.

Conclude

The function \(f(x) = x^2\) is not a periodic function since there is no non-zero value of T for which the function repeats itself.

Key concepts

Understanding Periodic Functions

In mathematics, a periodic function is one that repeats its values at regular intervals, known as the function's period. One of the simplest examples of a periodic function is the sine function, which repeats every interval of \(2\pi\). To determine whether a function is periodic, we must find if there exists a non-zero number \(T\), known as the fundamental period, such that the equation \(f(x + T) = f(x)\) holds true for all values of \(x\).

When analyzing a function for periodicity, it is important to verify the condition over its entire domain. If no such \(T\) exists, the function is classified as non-periodic, and it does not have a fundamental period. Understanding this concept is crucial when studying functions in trigonometry, signal processing, and various other applications in science and engineering.

Quadratic Functions and Their Properties

Quadratic functions are polynomial functions of degree two, generally expressed in the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These functions are graphically represented by parabolas, which are symmetric U-shaped curves. The vertex of the parabola marks the maximum or minimum point of the function, depending on the direction the parabola opens.

A key characteristic of quadratic functions is that they do not exhibit periodic behavior. Their graphs extend indefinitely without repeating patterns or intervals, as evidenced in the function \(f(x) = x^2\). Despite their non-periodic nature, quadratic functions are widely used to model phenomena in physics, economics, and many other fields, making them an essential element in the study of algebra.

Properties of Exponents

Exponents represent the power to which a number, known as the base, is raised. They are part of the fundamental tools in algebra and appear in various mathematical contexts. The properties of exponents dictate how to manipulate expressions involving powers and are essential for simplifying and solving equations.

For instance, when multiplying numbers with the same base, the exponents are added: \(a^m \cdot a^n = a^{m+n}\). Conversely, when dividing, the exponents are subtracted: \(\frac{a^m}{a^n} = a^{m-n}\). There's also the power of a power property: \((a^m)^n = a^{m \cdot n}\). These rules are indispensable when dealing with periodic functions because they aid in assessing whether a function's period could exist based on the transformation of the function's input by some period \(T\). In exploring periodicity in quadratic functions, we see that there isn't a consistent way to apply these properties to find a repeating interval, reinforcing the conclusion that quadratic functions are not periodic.