How to draw the graph of a function of $x$.

Question

$$y=\begin{cases} x^3 & x\le 1 \\ x & x \ge 1\end{cases}$$

Both are continuous at $x=1$. But not differentiate at the point.

Is the graph right?

Answer 1

Yes.

To the degree of detail. For $x\le1$ the graph will resemble the graph of $y = x^3$, and for $x > 1$ the graph will resemble the graph $y = x$. As at $x = 1$, both $x^3$ and $x$ equal $1$, the graph with "connect".

Answer 2

Here is a graph generated with MATHEMATICA

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Well, I have had installed the program in a PC. There are some sites where you can graph functions, but I'm not sure if you can get graphs with the same quality:

Wolfram Alpha

Desmos

Geogebra

I hope they will be useful to you.

Answer 3

The graph is correct, although the currently-attached statement $f'(x) = 3$ at $x \le 1$ is incorrect. It would be true to say that $f'(x) = 3x^2$ at $x < 1$, for which the limit of $f'(x)$ approaches 3 as $x$ approaches 1 from the left.