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Problem 42 Medium Difficulty

Find the numbers at which $ f $ is discontinuous. At which of these numbers is $ f $ continuous from the right, from the left, or neither? Sketch the graph of $ f $.

$ f(x) = \left\{
\begin{array}{ll}
2^x & \mbox{if $ x \le 1 $}\\
3 - x & \mbox{if $ 1 < x \le 4 $} \\
\sqrt{x} & \mbox{if $ x > 4 $}
\end{array} \right.$

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Video Transcript

Okay, so here we have the piecewise defined function F of X to graph a piecewise defined function in dez mose you write down what the function is, for example, F of X equals two to the ex. But since uh that's the definition of uh ffx only when in this case X is less than or equal to one. You have to put uh those particular X values. That dysfunction is only defined for these X values. You have to put it in these little curly brackets, that's how you put this in decimus. So the blue curve here is F of X equals to two D X for X less than or equal to one. The green portion of the graph of F of X. F of X is defined to be three minus X when X is between one and four in case that's the green portion of the grave, last but not least. The purple portion of the graph of F of X F of X is defined to be the square root of X when X is strictly greater than four. So for X greater than for the purple graph represents f of X equals the square root of X. Now you can see that this function is continuous almost everywhere, except when X equals four, X equals X equals for um f of X is discontinuous. And that's because as X approaches for from the negative side, the function is approaching negative one. And as X approaches for from the positive side, F of X approaches to. So we have a dis continuity at X equal sport uh The function F of X is continuous uh From the left of four, and function is also continuous from the right of X equal sport.