Question Number 205107 by mnjuly1970 last updated on 08/Mar/24
$$ \\ $$$$\:\:\:\:{y}\:=\:{log}_{\mathrm{2}} \left({sin}\left({x}\right)+{cos}\left({x}\right)\right) \\ $$$$\:\:\:\Rightarrow\:\:{R}_{{y}} \:=\:?\left({Range}\:\right) \\ $$$$ \\ $$
Commented by mr W last updated on 08/Mar/24
![2^(−∞) =0 < sin x+cos x ≤(√2)=2^(1/2) −∞< log_2 (sin x+cos x) ≤(1/2) i.e. R_y =(−∞, (1/2)]](https://www.tinkutara.com/question/Q205109.png)
$$\mathrm{2}^{−\infty} =\mathrm{0}\:<\:\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\:\leqslant\sqrt{\mathrm{2}}=\mathrm{2}^{\frac{\mathrm{1}}{\mathrm{2}}} \\ $$$$−\infty<\:\mathrm{log}_{\mathrm{2}} \:\left(\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)\:\leqslant\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${i}.{e}.\:{R}_{{y}} =\left(−\infty,\:\frac{\mathrm{1}}{\mathrm{2}}\right] \\ $$
Commented by mnjuly1970 last updated on 08/Mar/24
$$\:{grateful}\:\:{sir}\:{W} \\ $$