Question type 7: Determining whether the function has an inverse in the respective domains Exercises

Determine whether the function $f(x)=x^2$ with domain $x\ge0$ has an inverse. If it does, find the inverse function.

Determine whether the exponential function $f(x)=e^x$ on $\mathbb{R}$ has an inverse. If it does, find the inverse and state its domain.

Determine whether the natural logarithm function $f(x)=\ln(x)$ on $(0, \infty)$ has an inverse. If it does, find the inverse.

Determine whether the function $f(x)=x^2$ with domain $x<0$ has an inverse. If it does, find the inverse function.

Determine whether the sine function $f(x)=\sin(x)$ with domain $[-\tfrac{\pi}{2},\tfrac{\pi}{2}]$ has an inverse. If it does, find the inverse.

Determine whether the absolute value function $f(x)=|x|$ on $\mathbb{R}$ has an inverse. If not, suggest a domain restriction that makes it invertible and state the inverse.

Determine whether the quadratic function $f(x)=x^2-4x+5$ on domain $(-\infty,2]$ has an inverse. If it does, find the inverse.

Determine whether the tangent function $f(x)=\tan(x)$ on $(-\tfrac{\pi}{2},\tfrac{\pi}{2})$ has an inverse. If it does, find the inverse.

Determine whether the function $f(x)=x^4$ with domain $x\ge0$ has an inverse. If it does, find the inverse.

Determine whether the function $f(x)=x^3-2$ defined on all real numbers has an inverse. If it does, find the inverse.

Determine whether the function $f(x)=\sqrt{x+3}$ with domain $x\ge-3$ has an inverse. If it does, find the inverse.

Determine whether the reciprocal function $f(x)=\frac{1}{x}$ on domain $(0, \infty)$ has an inverse. If it does, find the inverse.