Continuity of Inverse Functions
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Problems 7
- The function f(x)=exf(x)=e^xf(x)=ex is continuous on R\mathbb{R}R. Find its inverse and the interval on which it is continuous.
- Is the inverse of f(x)={x2−4≤x≤0−x30<x≤4f(x)=\begin{cases}x^2&-4\leq x\leq0\\-x^3&0<x\leq4\end{cases}f(x)={x2−x3−4≤x≤00<x≤4 continuous?
- Is the inverse of f(x)={2x−4≤x≤0ln(x)0<x≤4f(x)=\begin{cases}2^x&-4\leq x\leq0\\\ln(x)&0<x\leq4\end{cases}f(x)={2xln(x)−4≤x≤00<x≤4 continuous?
- True or False\text{True or False}True or False === If f(x)f(x)f(x) is continuous on its domain then f−1f^{-1}f−1 is continuous on f(x)f(x)f(x)'s domain.
- Suppose f(x)=x3−6x2+12x+8f(x)=x^3-6x^2+12x+8f(x)=x3−6x2+12x+8 is continuous on the interval [−1,3][-1,3][−1,3]. What interval is its inverse continuous on?
- The function f(x)=ln(x)2ln(2)f(x)=\frac{\ln(x)}{2\ln(2)}f(x)=2ln(2)ln(x) is continuous on (0,∞)(0,\infty)(0,∞). Find its inverse and the interval on which it is continuous.
- Is the inverse of f(x)=1xf(x)=\frac1xf(x)=x1 continuous?