find the domain and range of the function f(x) = 2-3x^2.

also find f(-2) and the numbers which are associated with the numbers -25 in its range.


dear experts please explain me what the second part of the question means with the answer

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{Given} \\ \mathrm{f}\left(\mathrm{x}\right)=2-3{\mathrm{x}}^2 \\ \mathbf{Domain}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{the}\mathbf{ }\mathbf{function}\mathbf{ }\mathbf{will}\mathbf{ }\mathbf{be}\mathbf{ }\mathbf{x}\mathbf{\in }\mathbf{R} \\ \mathrm{To} \mathrm{find} \mathrm{range}: \\ \mathrm{f}\text{'}\left(\mathrm{x}\right)=-6\mathrm{x} \\ \mathrm{hence} \mathrm{for} \mathrm{x}<0, \mathrm{f}\text{'}\left(\mathrm{x}\right)>0---> \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{increasing} \mathrm{for} \mathrm{x}<0 \\ \mathrm{x}>0, \mathrm{f}\text{'}\left(\mathrm{x}\right)<0 ----> \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{decreasing} \mathrm{for} \mathrm{x}>0 \\ \mathrm{at} \mathrm{x}=0, \mathrm{f}\left(0\right)=2 \\ \mathrm{Drawing} \mathrm{a} \mathrm{graph} \mathrm{of} \mathrm{the} \mathrm{function}, \mathrm{it} \mathrm{will} \mathrm{be} \mathrm{an} \mathrm{inverted} \mathrm{parabola} \mathrm{as} \mathrm{shown} \mathrm{in} \mathrm{the} \mathrm{figure}. \\ \mathrm{Clearly} \mathrm{the} \mathrm{maximum} \mathrm{value} \mathrm{of} \mathrm{function} \mathrm{will} \mathrm{be} \mathrm{at} \mathrm{x}=0 \mathrm{which} \mathrm{is} 2 \\ \mathrm{and} \mathrm{minimum} \mathrm{value} \mathrm{will} \mathrm{be} -\infty \\ \mathbf{hence}\mathbf{ }\mathbf{range}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{f}\left(\mathbf{x}\right)\mathbf{=}\left(\mathbf{2}\mathbf{,}\mathbf{-}\mathbf{\infty }\right) \\ \mathrm{f}\left(-2\right)=2-3{\left(-2\right)}^2=2-12=-10 \\ \mathrm{f}\left(\mathrm{x}\right)=-25 \\ -25=2-3{\mathrm{x}}^2 \\ 3{\mathrm{x}}^2=27 \\ {\mathrm{x}}^2=9 \\ \mathrm{x}=\pm 3 \\ \mathbf{hence}\mathbf{ }\mathbf{numbers}\mathbf{ }\mathbf{associated}\mathbf{ }\mathbf{with}\mathbf{ }\mathbf{f}\left(\mathbf{x}\right)\mathbf{=}\mathbf{-}\mathbf{25}\mathbf{ }\mathbf{are}\mathbf{ }\mathbf{\pm }\mathbf{3} \\ \mathbf{this}\mathbf{ }\mathbf{simply}\mathbf{ }\mathbf{means}\mathbf{ }\mathbf{find}\mathbf{ }\mathbf{values}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{x}\mathbf{ }\mathbf{for}\mathbf{ }\mathbf{which}\mathbf{ }\mathbf{f}\left(\mathbf{x}\right)\mathbf{=}\mathbf{-}\mathbf{25} \end{gathered}$$

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