periodic function

Examples of periodic function in the following topics:

• Sine and Cosine as Functions

  • In the graphs for both sine and cosine functions, the shape of the graph repeats after $2\pi$, which means the functions are periodic with a period of $2\pi$.
  • A periodic function is a function with a repeated set of values at regular intervals.
  • When this occurs, we call the smallest such horizontal shift with $P>0$ the period of the function.
  • The diagram below shows several periods of the sine and cosine functions.
  • The sine and cosine functions are periodic, meaning that a specific horizontal shift, $P$, results in a function equal to the original function:$f(x + P) = f(x)$.

• Tangent as a Function

  • Characteristics of the tangent function can be observed in its graph.
  • The tangent function can be graphed by plotting $\left(x,f(x)\right)$ points.
  • As with the sine and cosine functions, tangent is a periodic function.
  • The period of the tangent function is $\pi$ because the graph repeats itself on $x$-axis intervals of $k\pi$, where $k$ is a constant.
  • The graph of the tangent function is symmetric around the origin, and thus is an odd function.

• Inverse Trigonometric Functions

  • Each trigonometric function has an inverse function that can be graphed.
  • To use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes” what the original trigonometric function “does,” as is the case with any other function and its inverse.
  • However, the sine, cosine, and tangent functions are not one-to-one functions.
  • In fact, no periodic function can be one-to-one because each output in its range corresponds to at least one input in every period, and there are an infinite number of periods.
  • As with other functions that are not one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one.

• Defining Trigonometric Functions on the Unit Circle

  • Identifying points on a unit circle allows one to apply trigonometric functions to any angle.
  • We have already defined the trigonometric functions in terms of right triangles.
  • The unit circle demonstrates the periodicity of trigonometric functions.
  • Periodicity refers to the way trigonometric functions result in a repeated set of values at regular intervals.
  • This is an indication of the periodicity of the cosine function.

• Polynomial and Rational Functions as Models

  • Functions are commonly used in fitting data to a trend line.
  • When we find a function that lies reasonably close to the collected data points, we create a trend line which says how one property behaves as a function of the other ones.
  • A rational function is the ratio of two polynomial functions and has the following form:
  • For example, if $n=2$ and $m=1$, the function is described as a quadratic/linear rational function.
  • Every time we say something through the phone, our phone tries to reduce the background noise by approximating our sound for short periods of time, again with the help of rational functions.

• Interest Compounded Continuously

  • Fundamentally, compound interest is an application of exponential functions that is found very commonly in every day life.
  • The amount of interest earned increases with each compounding period.
  • This can be seen by the fact that the amount after $n$ years is given by a linear function with slope equal to five (see table below).
  • The equation representing investment value as a function of principal, interest rate, period and time is:
  • The more frequent the compounding periods the more interest is accrued.

• Population Growth

  • Population can fluctuate positively or negatively and can be modeled using an exponential function.
  • Namely, it is given by the formula $P(r, t, f)=P_i(1+r)^\frac{t}{f}$ where $P{_i}$ represents the initial population, r is the rate of population growth (expressed as a decimal), t is elapsed time, and f is the period over which time population grows by a rate of r.
  • The rate $r$ by which the population is growing is itself a function of four variables.
  • If we multiply the $PGR$ by $100$ we arrive at the percentage growth relative to the population at the beginning of the time period.
  • If the current rates of births and deaths hold, the world population growth can be modeled using an exponential function.

• Introduction to Rational Functions

  • A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.
  • A rational function is any function which can be written as the ratio of two polynomial functions.
  • Any function of one variable, $x$, is called a rational function if, and only if, it can be written in the form:
  • Note that every polynomial function is a rational function with $Q(x) = 1$.
  • A constant function such as $f(x) = \pi$ is a rational function since constants are polynomials.

• Increasing, Decreasing, and Constant Functions

  • As part of exploring how functions change, we can identify intervals over which the function is changing in specific ways.
  • We say that a function is increasing on an interval if the function values increase as the input values increase within that interval.
  • In terms of a linear function $f(x)=mx+b$, if $m$ is positive, the function is increasing, if $m$ is negative, it is decreasing, and if $m$ is zero, the function is a constant function.
  • In mathematics, a constant function is a function whose values do not vary, regardless of the input into the function.  
  • A function is a constant function if $f(x)=c$ for all values of $x$ and some constant $c$.  

• Composition of Functions and Decomposing a Function

  • Functional composition allows for the application of one function to another; this step can be undone by using functional decomposition.
  • The process of combining functions so that the output of one function becomes the input of another is known as a composition of functions.
  • The resulting function is known as a composite function.
  • Note that the range of the inside function (the first function to be evaluated) needs to be within the domain of the outside function.
  • Practice functional composition by applying the rules of one function to the results of another function