a) A function f: R---->R is called periodic with period p if, for all x \"within
ID: 2984149 • Letter: A
Question
a) A function f: R---->R is called periodic with period p if, for all x "within" R it holds that f(x) = f(x+p). Prove that all continuous periodic functions are uniformly continuous and bounded.
b) Let f: [a,b]--->R be a continous injective function for which f(a) < f(b). Prove that f is strictly increasing.
Explanation / Answer
THIS WILL HELP YOU The fundamental period of a function is the period of the function which are of the form,f(x+k)=f(x), then k is called the period of the function and the function f is called a periodic function. Let us define the function h(t) on the interval [0,2] as follows: h(t)=???????????03t?11?3t+5 if x=0?t?1or if53?t?2 if x=13?t?23 if x=23?t?43 if x=43?t?53 If we extend the function h to all of R by the equation, h(t+2)=h(t) => h is periodic with the period 2. The graph of the function is shown below. Finding the Period of a FunctionBack to Top If a function repeats over at a constant period we say that is a periodic function. Basically it is represented like f(x) = f(x + p), p is the real number and this is the period of the function. Period means the time interval between the two occurrences of the wave. To find the period of the periodic function we have to use the following formula, Where Period = 2pb, here b is the co - efficient of x Sine and cosine functions have the forms of a periodic wave: Period: It is represented as "T", Period is the distance among two repeating points on the graph function. Amplitude: It is represented as "A" It is the distance between the middle point to highest or lowest point on the graph function. sin(a?) = 2?a and cos(a?) = 2?a Solved Examples Question 1: Find the period of the given periodic function. Where f(x) = 9sin(6px7 + 5) Solution: Given periodic function is f(x) = 9sin(6px7+ 5) To find the period we have the formulas period = 2pb Where period of the periodic function = 2p(6p7) = 146 = 73 Question 2: Find the period of the periodic function f(x) . Where f(x) = 9 Cos x Solution: The given periodic function is f(x) = 9 Cos x To find the period, we have the formula. period = 2?b Where period of the periodic function = 2p1 = 2p As we are aware that sin (2? + x) = sin x and cos (2? + x) = cos x.., we see that the periods of sine and cosine functions are 2?.. Also, tan (? + x) = tan x, hence the period of tan x is ? Let us graph the primary trigonometric functions. The following graph shows the function y = sin x Let us find the co-ordinates of the points to graph. x -2 ? -3 ?/2 - ? - ?/2 0 ?/2 ? 3 ?/2 2? 5 ?/2 y 1 1 0 -1 0 1 0 -1 0 1 Period = 2 ? Axis: y = 0 [x-axis ] Amplitude : 1 Maximum value = 1 Minimum value = -1 domain = { x : x R } Range { y : y R . -1 y 1 } The following graph shows the trigonometric function y = cos x Let us prepare the table to values x -2 ? -3 ?/2 - ? - ?/2 0 ?/2 ? 3 ?/2 2? 5 ?/2 y 1 0 -1 0 1 0 -1 0 1 0 Period: 2 ? Axis: y = 0 Amplitude = 1 Maximum value = 1 Minimum value = -1 Domain: { x : x R } Range { y : y R . -1 y 1 }
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