Explanation
We are given the function:
[tex]f(x)=-2.5sin(x-35^0)+1[/tex]We can start with the domain and range of the function:
The function given is a sine function
Domain of a function
[tex]\mathrm{The\:domain\:of\:a\:function\:is\:the\:set\:of\:input\:or\:argument\:values\:for\:which\:the\:function\:is\:real\:and\:defined}[/tex][tex]\begin{bmatrix}\mathrm{Solution:}\:&\:-\infty \:The range of the function[tex]\mathrm{The\:set\:of\:values\:of\:the\:dependent\:variable\:for\:which\:a\:function\:is\:defined}[/tex][tex]\begin{gathered} -1\le \sin \left(x-35^{\circ \:}\right)\le \:1 \\ -2.5\le \:-2.5\sin \left(x-35^{\circ \:}\right)\le \:2.5 \\ -1.5\le \:-2.5\sin \left(x-35^{\circ \:}\right)+1\le \:3.5 \\ \mathrm{Therefore\:the\:range\:is}:-1.5\le\:f\left(x\right)\le\:3.5 \end{gathered}[/tex]Thus, the range is
[tex]\begin{bmatrix}\mathrm{Solution:}\:&\:-1.5\le \:f\left(x\right)\le \:3.5\:\\ \:\mathrm{Interval\:Notation:}&\:\left[-1.5,\:3.5\right]\end{bmatrix}[/tex]Then we can now get the maximum and minimum values
[tex]\begin{gathered} \mathrm{Suppose\:that\:}x=c\mathrm{\:is\:a\:critical\:point\:of\:}f\left(x\right)\mathrm{\:then,\:} \\ \mathrm{If\:}f\:'\left(x\right)>0\mathrm{\:to\:the\:left\:of\:}x=c\mathrm{\:and\:}f\:'\left(x\right)<0\mathrm{\:to\:the\:right\:of\:}x=c\mathrm{\:then\:}x=c\mathrm{\:is\:a\:local\:maximum.} \\ \mathrm{If\:}f\:'\left(x\right)<0\mathrm{\:to\:the\:left\:of\:}x=c\mathrm{\:and\:}f\:'\left(x\right)>\:0\mathrm{\:to\:the\:right\:of\:}x=c\mathrm{\:then\:}x=c\mathrm{\:is\:a\:local\:minimum.} \\ \mathrm{If\:}f\:'\left(x\right)\mathrm{\:is\:the\:same\:sign\:on\:both\:sides\:of\:}x=c\mathrm{\:then\:}x=c\mathrm{\:is\:neither\:a\:local\:maximum\:nor\:a\:local\:minimum.} \end{gathered}[/tex]The critical points are
[tex]\begin{gathered} \mathrm{Plug\:the\:extreme\:point}\:x=-0.95993..+180^{\circ\:}+360^{\circ\:}n \\ \:\mathrm{into}\:-2.5\sin \left(x-35^{\circ \:}\right)+1\quad \Rightarrow \quad \:y=-1.5 \end{gathered}[/tex]So that the minimum will be
[tex]\mathrm{Minimum}\left(-0.95993...+180^{\circ\:}+360^{\circ\:}n,\:-1.5\right)[/tex]Also, the maximum is
[tex]\:\mathrm{Maximum}\left(5.32325...+360^{\circ\:}n,\:3.5\right)[/tex]