Consider the following definitions: Even Function : A function $f(x)$ is an even function if for e

Question

Consider the following definitions: Even Function : A function $f(x)$ is an even function if for every $x$ in its domain (the set of $x$ where the function is defined on), the function satisfies the following equation:$$f(x) = f(-x).$$ For example, $f(x) = x^2$ is an even function, since for every real number $x$, it satisfies that $f(x) = x^2 = (-x)^2 = f(-x)$. Odd Function : A function $f(x)$ is an odd function if for every $x$ in its domain, the function satisfies the following equation: $$f(x) = -f(-x).$$ For example, $f(x) = x^3$ is an odd function, since for every real number $x$, it satisfies that $f(x) = x^3 = -(-x)^3 = -f(-x)$. Periodic Function : A function $f(x)$ is a periodic function if there exists a positive number $T$ such that, for every $x$ in its domain, the function satisfies the following equation: $$f(x + T) = f(x),$$ where the smallest positive number $T$ is called the period of the function. For example, $f(x) = \sin x + 2$ is a periodic function, since for every real number $x$, there exists $T = 2\pi$ such that, $$f(x) = \sin x + 2 = \sin(x + 2\pi) + 2 = f(x + 2\pi),$$where $T = 2\pi$ is the period of $f(x)$. Using the definitions above, categorise the following functions: $y = \sin 5x$ $y = -4 \sin 2x + 3x$ $y = \cos(\sin x)$ $y = \tan 2x + 7x^5$ $y = \sin 3x \sin 50x$ $y = 1$ $y = 2 \cos 6x - 1$ $y = \cos 2x + x^2$ $y = \sin(\cos x)$ $y = \tan^2 3x$ $y = x \cos 20x$ $y = 0$ Even Function: Odd Function: Periodic Function:

Accepted Answer

Even: $2\cos6x-1$, $\cos2x+x^2$, $\cos(\sin x)$, $\sin(\cos x)$, $\tan^2 3x$, $\sin3x\sin50x$, $1$, $0$. Odd: $\sin5x$, $-4\sin2x+3x$, $\tan2x+7x^5$, $x\cos20x$, $0$. Periodic under the stated existence criterion: $\sin5x$, $2\cos6x-1$, $\cos(\sin x)$, $\sin(\cos x)$, $\tan^2 3x$, $\sin3x\sin50x$, $1$, $0$. The two constant functions have no least positive period.

Trigonometry - Trigonometric Graphs and Parameters

Trigonometry - Trigonometric Graphs and Parameters

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