Graphs of Trigonometric Functions

Trigonometric Graphs - Amplitude and Periodicity


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

Details and assumptions

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

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

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

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

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


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