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Category: Relation and Functions

let-f-x-x-4-x-2-and-g-x-2-x-3-2-x-3-1-find-D-f-D-g-and-D-fog-and-determine-fog-x-2-calculate-gof-x-and-give-D-gof-3-calculate-1-2-1-2-f-x-d

Question Number 59188 by maxmathsup by imad last updated on 05/May/19 $${let}\:{f}\left({x}\right)={x}−\sqrt{\mathrm{4}−{x}^{\mathrm{2}} }\:\:{and}\:{g}\left({x}\right)\:=\frac{\mathrm{2}\:+\sqrt{{x}−\mathrm{3}}}{\mathrm{2}−\sqrt{{x}−\mathrm{3}}} \\ $$$$\left.\mathrm{1}\right)\:{find}\:\:\:{D}_{{f}} \:\:,{D}_{{g}} \:\:\:{and}\:{D}_{{fog}} \:\:\:\:\:{and}\:\:{determine}\:{fog}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:{gof}\left({x}\right)\:{and}\:{give}\:{D}_{{gof}} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{−\frac{\mathrm{1}}{\mathrm{2}}} ^{\frac{\mathrm{1}}{\mathrm{2}}} {f}\left({x}\right){dx}\:\:\:\: \\…

If-f-2-x-3-2-4x-and-f-1-p-2-then-p-

Question Number 124444 by benjo_mathlover last updated on 03/Dec/20 $$\:{If}\:{f}\left(\frac{\mathrm{2}}{{x}}\right)\:=\:\frac{\mathrm{3}}{\mathrm{2}+\mathrm{4}{x}}\:{and}\:{f}^{−\mathrm{1}} \left({p}\right)=\mathrm{2}\: \\ $$$${then}\:{p}\:=? \\ $$ Answered by bemath last updated on 03/Dec/20 $$\:\Leftrightarrow\:{f}^{−\mathrm{1}} \left({p}\right)=\mathrm{2}\:{then}\:{f}\left(\mathrm{2}\right)={p} \\…

advanced-calculus-p-q-are-positive-integers-and-p-q-let-p-q-0-1-x-p-1-x-q-dx-prove-that-n-n-1-1-n-1-k-1-n-k-1-

Question Number 124102 by mnjuly1970 last updated on 30/Nov/20 $$\:\:\:\:\:\:\:\:\:\:\:…{advanced}\:\:\:{calculus}… \\ $$$$\:\:{p}\:,\:{q}\:{are}\:{positive}\:{integers}\:{and} \\ $$$${p}\geqslant{q}\:\::\:{let}\::\phi\left({p},{q}\right)=\int_{\mathrm{0}} ^{\:\mathrm{1}} {x}^{{p}} \left\{\frac{\mathrm{1}}{{x}}\right\}^{{q}} {dx} \\ $$$$\:\:{prove}\:{that}\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\phi\left({n},{n}\right)\overset{?} {=}\mathrm{1}−\frac{\mathrm{1}}{{n}+\mathrm{1}}\:\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\zeta\left({k}+\mathrm{1}\right)…