IB Mathematics Applications & Interpretation 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 $f^{-1}(x)$.