Question Type 6: Finding the range of more complex functions Exercises

Find the range of $f(x)=\dfrac{e^x}{1+e^{2x}}$ for $x\in\mathbb{R}$.

Find the range of $f(x)=2^{-|x|}$ for $x\in\mathbb{R}$.

Find the range of $f(x)=\dfrac{1}{x}+\dfrac{1}{x+2}$ for $x\in\mathbb{R}\setminus\{0,-2\}$.

Find the range of $f(x) = e^{1-x^2} + 2$ for $x \in \mathbb{R}$.

Find the range of $f(x)=\dfrac{\ln x}{x}$ for $x>0$.

Find the range of $f(x)=\ln\big(\sqrt{x^2+1}-x\big)$ for $x\in\mathbb{R}$.

Find the range of $f(x)=\tanh x=\dfrac{e^x-e^{-x}}{e^x+e^{-x}}$ for $x\in\mathbb{R}$.

Find the range of $f(x)=\dfrac{x^2-4x+5}{x^2+1}$ for $x\in\mathbb{R}$.

Find the range of $f(x)=\sin(\ln x)$ for $x>0$.

Find the range of $f(x)=\dfrac{x-1}{x^2+1}$ for $x\in\mathbb{R}$.

Find the range of the function $f(x)=2\cdot 3^{-\sqrt{x}}-1$ for $x\ge 0$.

Find the range of $f(x)=\dfrac{1}{\sqrt{2-\sin x}}$ for $x\in\mathbb{R}$.