Question Type 5: Finding the inverse of more complex equations and restrictions on the domain Bootcamps

Determine the inverse of $f(x) = \sqrt{\ln(x)}$, for $x \ge 1$.

Find the inverse of $f(x) = \ln\bigl((x - 4)^4\bigr)$, for $x>4$.

Find the inverse of $f(x) = \ln\bigl(\sqrt{x - 1}\bigr)$, for $x>1$.

Determine the inverse of $f(x) = \ln\bigl(5^x + 2\bigr)$.

Determine the inverse of $f(x) = \tfrac12\ln(5 - 3x)$, where $x<\tfrac{5}{3}$.

Find $f^{-1}(x)$ if $f(x) = \sqrt{5 - \ln(2x)}$, $0< x \le \tfrac{e^5}{2}$.

Determine the inverse of $f(x) = e^{(x+2)^3} - 1$.

Find $f^{-1}(x)$ for $f(x) = \sqrt{\ln(x^2 + 1)}$, where $x\ge0$.

Find the inverse of $f(x) = 10\sqrt{\ln(7x^3 - 3)}$, where $x \ge (\tfrac{4}{7})^{1/3}$.

Find $f^{-1}(x)$ if $f(x)=3\sqrt[5]{\ln\bigl(\tfrac{x+1}{2}\bigr)}$, for $x>-1$.

Determine the inverse of $f(x) = \ln\bigl(\tfrac{2x + 1}{3 - x}\bigr)$, for $x<3$.

Find the inverse of $f(x) = \dfrac{\ln(x) + 1}{\ln(x) - 1}$, for $x>1$, $x\neq e$.