Mathematics - Functions Question with Solution | TestHub

We must have

$$\begin{array}{rl}{9}^x+27^{\frac{2}{3}(x-2)}-219-3^{2(x-1)}& \ge 0\\ {\left(3^x\right)}^2+3^{2(x-2)}-219-3^{2x-2}& \ge 0\\ \left(3^{2x}\right)+\frac{3^{2x}}{81}-219-\frac{3^{2x}}{9}& \ge 0\\ \left(1+\frac{1}{81}-\frac{1}{9}\right)3^{2x}& \ge 219\\ \frac{73}{81}\times 3^{2x}& \ge 219\\ 3^{2x}& \ge 3\times 81=3^5\\ 2x& \ge 5\\ x& \ge \frac{5}{2}\\ \Rightarrow \text{ Domain is }\left[\frac{5}{2},\infty \right).& \end{array}$$