栵 1 Find the monotonic intervals and range of the function $y=3^{-x^{2}+2 x+3}$.


Let $u=-x^{2}+2 x+3, y=3^{u}$.
Since $y=3^{u}$ is an increasing function on $\mathbf{R}$, and $u=-x^{2}+2 x+3=-(x-1)^{2}+4$
is an increasing function on $(-\infty, 1]$ and a decreasing function on $[1,+\infty)$, therefore $y=f(x)$ is an increasing function on $(-\infty, 1]$ and a decreasing function on $[1,+\infty)$.
When $x=1$, $y_{\text {max }}=f(1)=81$, and since $y=3^{-x^{2}+2 x+3}>0$, the range of the function is $(0,81]$.
Key Points for Solving Generally, for a composite function $f[g(x)]$, its monotonicity is determined by the monotonicity of the functions $u=g(x)$ and $y=f(u)$. When the monotonicity of $u=g(x)$ and $y=f(u)$ is consistent, the function $f[g(x)]$ is increasing on its domain; when the monotonicity of $u=g(x)$ and $y=f(u)$ is opposite, the function $f[g(x)]$ is decreasing on its domain.