Which of the following functions is $$one-one$$?

Question

Infinity Learn · Accepted Answer

(a) $$f(x)=esgn$$⁡$$x+ex2$$ when $$x=0$$ when x>$$0$$ then $$f(0)=2$$ then $$f(x)=e+ex2$$ when x<$$0$$  then $$f(x)=1e+ex2$$ (b) We have $$f(x)=ex2+$$|x|,x∈[−$$1$$,∞$$)$$ clearly $$f($$−$$1)=e2=f(1)$$ also  $$x2+$$|x|≥$$0$$∀x∈[−$$1$$,∞]⇒ $$Rf=$$[$$1$$,∞$$)$$∴$$f(x)$$ is $$many-one$$ into function.  (c) $$f(x)=$$∣x−$$1$$|$$+$$|x−$$2$$|$$+$$|x−$$3$$|$$+$$|x−$$4$$∣,x∈[$$3$$,$$4$$]∴ $$f(x)=(x$$−$$1)+(x$$−$$2)+(x$$−$$3)$$−(x$$−$$4)=(3x$$−$$6)−(x$$−$$4)f(x)=2x$$−$$2$$, which is increasing function $$Rf=$$[$$4$$,$$6$$] Clearly, $$f(x) is $$one-one$$ onto function.  (d) $$f(x)=ln$$⁡$$($$$$cos$$⁡$$($$$$sin$$⁡$$x))$$ For domain, ln⁡$$($$$$cos$$⁡$$($$$$sin$$⁡$$x))$$≥$$0$$⇒    $$($$$$cos$$⁡$$($$$$sin$$⁡$$x))$$≥$$1$$⇒    $$cos$$⁡$$($$$$sin$$⁡$$x)=1$$∴    $$sin$$⁡$$x=0$$⇒     $$x=n$$π,n∈I    $$Rf={0}$$ since $$f(x)=0$$ Thus, $$f(x)$$ is $$many-one$$ function.

Which of the following functions is one-one?

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