The range of a function y $$=$$ f (x) is the set of all possible output values f (x) corresponding to every input x in the domain of f and is denoted as f (A) if A is the domain. For finding the range of a function y $$=$$ f (x), first of all, find the domain of f.If the domain consists of finite number of points, then the range consists of set of corresponding f (x) values. If the domain consists of whole real line or real line minus some finite points, then express x in terms of y as x $$=$$ $$g(y)$$. The values of y for which g is defined is the required range.If the domain is a finite interval, find the intervals in which f (x) $$increases/decreases$$ and then find the extreme values of the function in those intervals. The union of those intervals is the required range.If ex $$+$$ $$ef(x)$$ $$=$$ e, then range of the function f isThe range of the function $$f(x)=$$$$sin$$⁡π$$x2+1x4+1$$. where [⋅] denotes the greatest integer function, isRange of values of f (x) $$=$$ $$1$$ $$+$$ sinx $$+$$ $$sin3x$$ $$+$$ $$sin5x+$$….., x ∈ −π$$2$$,π$$2$$ is

Question

The range of a function y $$=$$ f (x) is the set of all possible output values f (x) corresponding to every input x in the domain of f and is denoted as f (A) if A is the domain. For finding the range of a function y $$=$$ f (x), first of all, find the domain of f.If the domain consists of finite number of points, then the range consists of set of corresponding f (x) values. If the domain consists of whole real line or real line minus some finite points, then express x in terms of y as x $$=$$ $$g(y)$$. The values of y for which g is defined is the required range.If the domain is a finite interval, find the intervals in which f (x) $$increases/decreases$$ and then find the extreme values of the function in those intervals. The union of those intervals is the required range.If ex $$+$$ $$ef(x)$$ $$=$$ e, then range of the function f isThe range of the function $$f(x)=$$$$sin$$⁡π$$x2+1x4+1$$. where [⋅] denotes the greatest integer function, isRange of values of f (x) $$=$$ $$1$$ $$+$$ sinx $$+$$ $$sin3x$$ $$+$$ $$sin5x+$$….., x ∈ −π$$2$$,π$$2$$ is

Infinity Learn · Accepted Answer

We have,ex $$+$$ $$ef(x)$$ $$=$$ e ⇒ $$ef(x)$$ $$=$$ e – ex⇒ f (x) $$=$$ log (e$$ – $$ex)For$$ f $$(x) to be defined, e – ex > $$0$$⇒ $$e1$$ > ex ⇒ x < $$1$$∴ Domain of f $$=$$ $$($$– ∞, $$1)Let$$ y $$=$$ log (e$$ – $$ex) ⇒ ey $$=$$ e – ex⇒ ex $$=$$ e – ey⇒ x $$=$$ log (e$$ – $$ey)For$$ x to be real, e – ey > $$0$$⇒ $$e1$$ > ey ⇒ y < $$1$$∴ Range of f $$=$$ $$($$– ∞, $$1)Since$$ [$$x2$$ $$+$$ $$1$$] is an integer,∴$$sin$$⁡π$$x2+1=0$$⇒$$f(x)=$$$$sin$$⁡π$$x2+1x4+1=0Hence$$, Range of f $$=$$ Rf $$=$$ $${0}We$$ have, $$f(x)=1+$$$$sin$$⁡$$xcos2$$⁡x⇒f′$$(x)=cos2$$⁡$$x($$$$cos$$⁡$$x)+$$$$sin$$⁡$$x(2cos$$⁡xsin⁡$$x)cos4$$⁡$$x=$$$$cos$$⁡$$xcos2$$⁡$$x+2sin2$$⁡$$xcos4$$⁡$$x=1+sin2$$⁡$$xcos3$$⁡x⇒ f ′$$(x) > $$0$$.∴ f (x) is increasing function.limx→−π$$2$$ $$1+$$$$sin$$⁡$$xcos2$$⁡$$x=$$−∞ and, limx→π$$2$$ $$1+$$$$sin$$⁡$$xcos2$$⁡$$x=$$∞∴ Range $$=$$ $$($$–∞, ∞$$)$$

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