IB Mathematics Analysis and Approaches Question Type 6: Finding the Range of More Complex Functions Exercises

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

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

Find the range of $f(x)=\frac{1}{x}+\frac{1}{x+2}$ 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 $f(x)=\frac{\ln x}{x}$ for $x>0$.

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

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

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

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

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