The domine of definition of the function
1 answer:
![\large\underline{\sf{Solution-}}](https://tex.z-dn.net/?f=%5Clarge%5Cunderline%7B%5Csf%7BSolution-%7D%7D)
<u>Given function is </u>
![\rm \longmapsto\:f(x) = \dfrac{1}{ \sqrt{ |cosx| + cosx} }](https://tex.z-dn.net/?f=%5Crm%20%5Clongmapsto%5C%3Af%28x%29%20%3D%20%5Cdfrac%7B1%7D%7B%20%5Csqrt%7B%20%7Ccosx%7C%20%20%2B%20cosx%7D%20%7D%20)
<u>Now, </u>
![\rm \longmapsto\:f(x) \: is \: defined \: if \: |cosx| + cosx > 0](https://tex.z-dn.net/?f=%5Crm%20%5Clongmapsto%5C%3Af%28x%29%20%5C%3A%20is%20%5C%3A%20defined%20%5C%3A%20if%20%5C%3A%20%20%7Ccosx%7C%20%2B%20cosx%20%3E%200)
We know,
![\begin{gathered}\begin{gathered}\bf\: \rm \longmapsto\: |x| = \begin{cases} &\sf{ - x \: \: when \: x < 0} \\ \\ &\sf{ \: \: x \: \: when \: x \geqslant 0} \end{cases}\end{gathered}\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5Cbegin%7Bgathered%7D%5Cbf%5C%3A%20%5Crm%20%5Clongmapsto%5C%3A%20%7Cx%7C%20%20%3D%20%5Cbegin%7Bcases%7D%20%26%5Csf%7B%20-%20x%20%5C%3A%20%20%5C%3A%20when%20%5C%3A%20x%20%3C%200%7D%20%5C%5C%20%20%5C%5C%20%26%5Csf%7B%20%5C%3A%20%20%5C%3A%20x%20%5C%3A%20%20%5C%3A%20when%20%5C%3A%20x%20%5Cgeqslant%200%7D%20%5Cend%7Bcases%7D%5Cend%7Bgathered%7D%5Cend%7Bgathered%7D)
So,
![\begin{gathered}\begin{gathered}\bf\: \rm \longmapsto\: |cosx| + cosx = \begin{cases} &\sf{ \: \: 0 \: \: when \: cosx \leqslant 0} \\ \\ &\sf{ \: \: 2 \: cosx \: \: when \: cosx > 0} \end{cases}\end{gathered}\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5Cbegin%7Bgathered%7D%5Cbf%5C%3A%20%5Crm%20%5Clongmapsto%5C%3A%20%7Ccosx%7C%20%2B%20cosx%20%20%3D%20%5Cbegin%7Bcases%7D%20%26%5Csf%7B%20%20%5C%3A%20%20%5C%3A%200%20%5C%3A%20%20%5C%3A%20when%20%5C%3A%20cosx%20%20%5Cleqslant%200%7D%20%5C%5C%20%20%5C%5C%20%26%5Csf%7B%20%5C%3A%20%20%5C%3A%202%20%5C%3A%20cosx%20%5C%3A%20%20%5C%3A%20when%20%5C%3A%20cosx%20%20%3E%20%200%7D%20%5Cend%7Bcases%7D%5Cend%7Bgathered%7D%5Cend%7Bgathered%7D)
So,
![\rm\implies \:f(x) \: is \: defined \: when \: cosx > 0](https://tex.z-dn.net/?f=%5Crm%5Cimplies%20%5C%3Af%28x%29%20%5C%3A%20is%20%5C%3A%20defined%20%5C%3A%20when%20%5C%3A%20cosx%20%3E%200)
So, from graph we concluded that cosx > 0 in the following intervals.
![\begin{gathered}\boxed{\begin{array}{c|c} \bf cosx & \bf \: x \: \in \: \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf + ve & \sf \bigg( - \dfrac{\pi}{2} ,\dfrac{\pi}{2} \bigg) \\ \\ \sf + ve & \sf \bigg(\dfrac{3\pi}{2} ,\dfrac{5\pi}{2} \bigg) \\ \\ \sf + ve & \sf \bigg(\dfrac{7\pi}{2} ,\dfrac{9\pi}{2} \bigg) \end{array}} \\ \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5Cboxed%7B%5Cbegin%7Barray%7D%7Bc%7Cc%7D%20%5Cbf%20cosx%20%26%20%5Cbf%20%5C%3A%20x%20%5C%3A%20%20%5Cin%20%5C%3A%20%20%5C%5C%20%5Cfrac%7B%5Cqquad%20%5Cqquad%7D%7B%7D%20%26%20%5Cfrac%7B%5Cqquad%20%5Cqquad%7D%7B%7D%20%5C%5C%20%5Csf%20%20%2B%20ve%20%26%20%5Csf%20%5Cbigg%28%20-%20%5Cdfrac%7B%5Cpi%7D%7B2%7D%20%2C%5Cdfrac%7B%5Cpi%7D%7B2%7D%20%5Cbigg%29%20%20%5C%5C%20%5C%5C%20%5Csf%20%20%2B%20ve%20%26%20%5Csf%20%5Cbigg%28%5Cdfrac%7B3%5Cpi%7D%7B2%7D%20%2C%5Cdfrac%7B5%5Cpi%7D%7B2%7D%20%5Cbigg%29%20%5C%5C%20%5C%5C%20%5Csf%20%20%2B%20ve%20%26%20%5Csf%20%5Cbigg%28%5Cdfrac%7B7%5Cpi%7D%7B2%7D%20%2C%5Cdfrac%7B9%5Cpi%7D%7B2%7D%20%5Cbigg%29%20%5Cend%7Barray%7D%7D%20%5C%5C%20%5Cend%7Bgathered%7D)
So, if we generalized this we get
![\rm\implies \:cosx > 0 \: when \: x \: \in \: \bigg(\dfrac{(4n - 1)\pi}{2} ,\dfrac{(4n + 1)\pi}{2} \bigg) \: \forall \: n \in \: Z](https://tex.z-dn.net/?f=%5Crm%5Cimplies%20%5C%3Acosx%20%3E%200%20%5C%3A%20when%20%5C%3A%20x%20%5C%3A%20%20%5Cin%20%5C%3A%20%5Cbigg%28%5Cdfrac%7B%284n%20-%201%29%5Cpi%7D%7B2%7D%20%2C%5Cdfrac%7B%284n%20%2B%201%29%5Cpi%7D%7B2%7D%20%5Cbigg%29%20%5C%3A%20%20%5Cforall%20%5C%3A%20n%20%5Cin%20%5C%3A%20Z)
<u>Hence, </u>
Domain of the function is
![\red{\rm\implies \boxed{\tt{ \: x \in \: \bigg(\dfrac{(4n - 1)\pi}{2} ,\dfrac{(4n + 1)\pi}{2} \bigg) \: \forall \: n \in \: Z}}}](https://tex.z-dn.net/?f=%5Cred%7B%5Crm%5Cimplies%20%5Cboxed%7B%5Ctt%7B%20%5C%3A%20x%20%5Cin%20%5C%3A%20%5Cbigg%28%5Cdfrac%7B%284n%20-%201%29%5Cpi%7D%7B2%7D%20%2C%5Cdfrac%7B%284n%20%2B%201%29%5Cpi%7D%7B2%7D%20%5Cbigg%29%20%5C%3A%20%20%5Cforall%20%5C%3A%20n%20%5Cin%20%5C%3A%20Z%7D%7D%7D)
![\textsf{More to know :-} \\](https://tex.z-dn.net/?f=%5Ctextsf%7BMore%20to%20know%20%3A-%7D%20%5C%5C)
![\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf T-eq & \bf Solution \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf sinx = 0 & \sf x = n\pi \: \forall \: n \in \: Z\\ \\ \sf cosx = 0 & \sf x = (2n + 1)\dfrac{\pi}{2}\: \forall \: n \in \: Z\\ \\ \sf tanx = 0 & \sf x = n\pi\: \forall \: n \in \: Z\\ \\ \sf sinx = siny & \sf x = n\pi + {( - 1)}^{n}y \: \forall \: n \in \: Z\\ \\ \sf cosx = cosy & \sf x = 2n\pi \pm \: y\: \forall \: n \in \: Z\\ \\ \sf tanx = tany & \sf x = n\pi + y \: \forall \: n \in \: Z\end{array}} \\ \end{gathered}\end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%5Cbegin%7Bgathered%7D%5Cboxed%7B%5Cbegin%7Barray%7D%7Bc%7Cc%7D%20%5Cbf%20T-eq%20%26%20%5Cbf%20Solution%20%5C%5C%20%5Cfrac%7B%5Cqquad%20%5Cqquad%7D%7B%7D%20%26%20%5Cfrac%7B%5Cqquad%20%5Cqquad%7D%7B%7D%20%5C%5C%20%5Csf%20sinx%20%3D%200%20%26%20%5Csf%20x%20%3D%20n%5Cpi%20%20%5C%3A%20%5Cforall%20%5C%3A%20n%20%5Cin%20%5C%3A%20Z%5C%5C%20%5C%5C%20%5Csf%20cosx%20%3D%200%20%26%20%5Csf%20x%20%3D%20%282n%20%2B%201%29%5Cdfrac%7B%5Cpi%7D%7B2%7D%5C%3A%20%5Cforall%20%5C%3A%20n%20%5Cin%20%5C%3A%20Z%5C%5C%20%5C%5C%20%5Csf%20tanx%20%3D%200%20%26%20%5Csf%20x%20%3D%20n%5Cpi%5C%3A%20%5Cforall%20%5C%3A%20n%20%5Cin%20%5C%3A%20Z%5C%5C%20%5C%5C%20%5Csf%20sinx%20%3D%20siny%20%26%20%5Csf%20x%20%3D%20n%5Cpi%20%2B%20%7B%28%20-%201%29%7D%5E%7Bn%7Dy%20%5C%3A%20%5Cforall%20%5C%3A%20n%20%5Cin%20%5C%3A%20Z%5C%5C%20%5C%5C%20%5Csf%20cosx%20%3D%20cosy%20%26%20%5Csf%20x%20%3D%202n%5Cpi%20%5Cpm%20%5C%3A%20y%5C%3A%20%5Cforall%20%5C%3A%20n%20%5Cin%20%5C%3A%20Z%5C%5C%20%5C%5C%20%5Csf%20tanx%20%3D%20tany%20%26%20%5Csf%20x%20%3D%20n%5Cpi%20%2B%20y%20%5C%3A%20%5Cforall%20%5C%3A%20n%20%5Cin%20%5C%3A%20Z%5Cend%7Barray%7D%7D%20%5C%5C%20%5Cend%7Bgathered%7D%5Cend%7Bgathered%7D)
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