The function \(f(x)=\operatorname{sech}(x)\) on \(\mathbf{R}\) has the range
The function \(f(x)=\operatorname{sech}(x)\) on \(\mathbf{R}\) has the range
\((0, \infty)\)
\((0,1]\)
\([1, \infty)\)
\((1, \infty)\)
Solution
\(f(x)=\operatorname{sech} x\)
\(=\frac{1}{\cosh x}=\frac{2}{e^x+e^{-x}}\)
At \(\quad x=0, f(x)=\frac{2}{e^0+e^{-0}}=\frac{2}{2}=1\)
At \(x \neq 0, f(x)\) is less than 1 or \(e^x+e^{-x}\) is greater than 2 .
Also \(f(x) \neq 0\) at any value of \(x\).
Hence, range \(\operatorname{sech} x\)
\(=(0,1]\)