11.12 Find the domain of the function $y=\sqrt{2^{x}-3^{x}}$.

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11.12 I method. The function is defined under the condition $2^{x}-3^{x} \geq 0$, i.e., $2^{x} \geq 3^{x}$. Dividing both sides of the inequality by $3^{x}>0$, we get $\left(\frac{2}{3}\right)^{x} \geq 1$, from which we obtain $x \leq 0$.

II method. On the same diagram, we plot the graphs of the functions $\quad y=2^{x} \quad$ and

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-349.jpg?height=479&width=439&top_left_y=630&top_left_x=692)

$y=3^{x}$ (Fig. 11.7).

Using the diagram, we get the answer: $(-\infty, 0]$.

Answer: $(-\infty, 0]$.