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

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

Find $f^{-1}(x)$ for $f(x) = \sqrt{\ln(x^2 + 1)}$, where $x \ge 0$.

Determine the inverse of $f(x) = \frac{1}{2}\ln(5 - 3x)$, where $x < \frac{5}{3}$.

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

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

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

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

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

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

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

Determine the inverse of $f(x) = \ln\left(\frac{2x + 1}{3 - x}\right)$, for $x < 3$.

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