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

Find the range of the quadratic function $k(x)=(x-2)^2+3$ for all real $x$.

Find the range of the absolute‐value function $t(x)=|x|+2$ for all real $x$.

Find the range of the inverse tangent function $r(x)=\arctan(x)$ for all real $x$.

Determine the range of $h(x)=-3\sqrt{x+2}+2$ for $x\ge-2$.

Find the range of $g(x)=\ln(x-1)+4$ given $x>1$.

Determine the range of $m(x)=\frac{4}{x+1}-2$ for $x\neq -1$.

Determine the range of the hyperbolic sine function $s(x)=\sinh(x)$ for all real $x$.

Find the range of $p(x)=7-2\cdot5^x$ for all real $x$.

Determine the range of $f(x)=5e^{-2x}+1$ for all real $x$.

Determine the range of $q(x)=3^{\sin x}$ for real $x$.

Find the range of the function $f(x)=2\cdot3^{-\sqrt{x}}-1$ for $x\ge0$.

Determine the range of $u(x)=2\cdot3^{-\sqrt{x}}-1$ when $1\le x\le4$.