Graphs of Trigonometric Functions

Trigonometric Graphs - Amplitude and Periodicity


Consider two functions f(x)=25sin6x,g(x)=x2.f(x)=25-\sin 6x, g(x)=\left\lfloor \frac{x}{2} \right\rfloor. Find the sum of all the elements in the range of g(f(x))g(f(x)).

Details and assumptions

The function x:RZ\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z} refers to the greatest integer smaller than or equal to xx. For example, 2.3=2\lfloor 2.3 \rfloor = 2 and 5=5\lfloor -5 \rfloor = -5.

If the period of the function f(x)=224+cosx+226cosxf(x)=\sqrt{224+\cos x}+\sqrt{226-\cos x} is aa, what is the value of f(a)f(a)?

Consider two functions f(x)=7sinx+12,g(x)=πcosπx.f(x)=7\sin x+12, g(x)=\pi\cos \pi x. If aa and bb are the maximum and minimum values, respectively, of f(g(x))f(g(x)), what is a+ba+b?

The graph of y=7tan(2πxπ)+18y=7 \tan (2\pi x-\pi)+18 has period 1a\frac{1}{a}. What is the value of aa?

What is the sum of the maximum and minimum values of the function y=sinx7+20?y=|\sin x-7|+20?


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