Find whether the following function is one one or many one and into or many one. (3x²/4π) - cos(πx)

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$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\frac{3{\mathrm{x}}^2}{4\mathrm{\pi }}-\cos \left(\mathrm{\pi x}\right)...\left(\mathrm{i}\right) \\ \mathrm{f}\left(-\mathrm{x}\right)=\frac{3{\left(-\mathrm{x}\right)}^2}{4\mathrm{\pi }}-\cos \left(-\mathrm{\pi x}\right) \\ \mathrm{Use} \cos \left(-\mathrm{A}\right)=\mathrm{cosA} \\ \Rightarrow \mathrm{f}\left(-\mathrm{x}\right)=\frac{3{\mathrm{x}}^2}{4\mathrm{\pi }}-\cos \left(\mathrm{\pi x}\right)...\left(\mathrm{ii}\right) \\ \mathrm{By} \left(\mathrm{i}\right) \mathrm{and} \left(\mathrm{ii}\right) \\ \mathrm{f}\left(-\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right) \\ \mathrm{Hence} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{an} \mathrm{even} \mathrm{function} \mathrm{and} \mathrm{all} \mathrm{even} \mathrm{functions} \mathrm{are} \mathrm{many} \mathrm{one}. \\ \mathrm{Hence} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{many}-\mathrm{one}. \\ \mathrm{Now} \mathrm{cos\theta }\in \left[-1,1\right] \mathrm{and} \frac{3{\mathrm{x}}^2}{4\mathrm{\pi }}\ge 0 \mathrm{for} \mathrm{all} \mathrm{x}\in \mathrm{R}, \mathrm{hence} \\ \mathrm{f}\left(\mathrm{x}\right) \mathrm{will} \mathrm{not} \mathrm{be} \mathrm{able} \mathrm{to} \mathrm{achieve} \mathrm{large} \mathrm{negative} \mathrm{values} \mathrm{which} \mathrm{means} \\ \mathrm{range} \mathrm{of} \mathrm{f}\left(\mathrm{x}\right) \mathrm{will} \mathrm{not} \mathrm{be} \mathrm{R}. \\ \mathrm{Hence} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{into} \mathrm{function}. \\ \mathrm{Hence} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is} \mathrm{many}-\mathrm{one} \mathrm{and} \mathrm{into} \mathrm{function}. \end{gathered}$$

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