Question Number 165808 by ZiYangLee last updated on 08/Feb/22
$$\mathrm{Which}\:\mathrm{of}\:\mathrm{the}\:\mathrm{following}\:\mathrm{is}\:\mathrm{an}\:\mathrm{even}\:\mathrm{function}? \\ $$$$\mathrm{A}.\:{f}_{\mathrm{1}} \left({x}\right)=\:\frac{\mathrm{sin}\:{x}}{\mathrm{3}^{{x}} +\mathrm{3}^{−{x}} }\:\:\:\:\:\:\:\:\:\mathrm{B}.\:{f}_{\mathrm{2}} \left({x}\right)=\:\frac{\mathrm{cos}\:{x}}{\mathrm{3}^{{x}} +\mathrm{3}^{−{x}} } \\ $$$$\mathrm{C}.\:{f}_{\mathrm{3}} \left({x}\right)=\mathrm{log}_{\mathrm{10}} \left({x}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}\right) \\ $$$$\mathrm{D}.\:{f}_{\mathrm{4}} \left({x}\right)=\:\frac{{x}^{\mathrm{2}} }{\mathrm{10}^{{x}} −\mathrm{1}} \\ $$
Answered by Rasheed.Sindhi last updated on 08/Feb/22
$$\mathrm{B}.\:\:{f}_{\mathrm{2}} \left(−{x}\right)=\frac{\mathrm{cos}\left(−{x}\right)\:}{\mathrm{3}^{−{x}} +\mathrm{3}^{−\left(−{x}\right)} }=\frac{\mathrm{cos}\:{x}}{\mathrm{3}^{{x}} +\mathrm{3}^{−{x}} }={f}_{\mathrm{2}} \left({x}\right) \\ $$$$\therefore\:{f}_{\mathrm{2}} \:{is}\:{an}\:{even}\:{function}. \\ $$$$ \\ $$