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Period of a Function

Tool to compute the period of a function: the value t such that the function repeats itself: f(x+t)=f(x-t)=f(x), that is the case for trigo functions (cos, sin, etc.)

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Period of a Function -

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# Period of a Function

## Answers to Questions (FAQ)

### What is the period of a function? (Definition)

The period $t$ of a periodic function $f(x)$ is the smallest value $t$ such that $$f(x+t) = f(x)$$

Graphically, its curve is repeated over the interval of each period. The function is equal to itself for every cycle of length $t$ (it presents a pattern/graph that is repeated by translation).

The value of the period $t$ is also called the periodicity of the function or fundamental period.

### How to find the period of a function?

To find the period $t$ of a signal or a function $f(x)$, demonstrate that $$f(x+t)=f(x)$$

Example: The trigonometric function $\sin(x + 2\pi) = \sin(x)$ so $\sin(x)$ is periodic of period $2\pi$ Trigonometric/sinusoidal functions are usually periodic, with a period $2\pi$, to guess the period, try multiples of pi for value $t$.

If the period is equal to 0, then the function is not periodic.

### How to find the value f(x) of a periodic function?

Any periodic function of period $t$ repeats every $t$ values. To predict the cycle value of a periodic function, for a value $x$ calculate $x_t = x \mod t$ (modulo t) and find the known value of $f(x_t) = f(x)$

Example: The function $f(x) = \cos(x)$ has a period of $2\pi$, the value for $x = 9 \pi$ is the same as for $x \equiv 9 \pi \mod 2 \pi \equiv \pi \mod 2 \pi$ and therefore $\cos(9 \pi) = \cos(\pi) = -1$

### How to find the amplitude of a periodic function?

The amplitude is the absolute value of the non-periodic part of the function.

Example: $a \sin(x)$ has for amplitude $| a |$

### How to prove the periodicity of a function?

The demonstration of the existence of a period $t$ for a function $f$ consists in calculating if the equation $f(x+t)=f(x)$ is true.

### How to prove that a function is not periodic?

If $f$ is periodic, then it exists a real not null such as $$f(x+t)=f(x)$$

Demonstration consists in proving that it is impossible. For example with a reductio ad absurdum or performing a calculation that leads to a contradiction.

### What are usual periodic functions?

The most common periodic functions are trigonometric functions based on sine and cosine functions (which have a period of 2 Pi).

FunctionPeriod
Sine $\sin(x) $$2\pi Cosine \cos(x)$$ 2\pi$
Tangent $\tan(x)$$\pi$

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