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

Determine whether the cubic function $f(x)=x^3-x$ with domain $(-\infty,\infty)$ has an inverse. Justify your answer.

Determine whether the function $f(x) = x^3$ with domain $(-\infty, \infty)$ has an inverse. If it does, find the inverse.

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

Determine whether the sine function $f(x) = \sin(x)$ with domain $(-\infty, \infty)$ has an inverse. Justify your answer.

Determine whether the function $f(x) = x^2$ with domain $(-\infty, \infty)$ has an inverse. Explain your reasoning.

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

Determine whether the exponential function $f(x) = e^x$ with domain $(-\infty, \infty)$ has an inverse. If so, find it.

Determine whether the tangent function $f(x)=\tan(x)$ with domain $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ has an inverse. If yes, find it.

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

Determine whether the quadratic function $f(x)=x^2+2x+1$ with domain $(-\infty,\infty)$ has an inverse. Explain.

Determine whether the function $f(x) = x^2$ with domain $(-\infty,0]$ has an inverse. If so, find its expression.

Determine whether the function $f(x)=x^2+2x+1$ restricted to domain $[-1,\infty)$ has an inverse. If so, find $f^{-1}(x)$.