Answer
The graph of the function is shown below.
Work Step by Step
$$y=x^{\frac{2}{3}}+\frac{1}{x^{\frac{1}{3}}}$$
The graph is shown below (the red line).
- From the formula, we see that the formula is the combination of two smaller functions $x^{2/3}$ and $1/(x^{1/3})$, which corresponds to two dominant terms of the function.
- As $x\to0$, $x^{2/3}$ approaches $0$ while $1/(x^{1/3})$ approaches $\pm\infty$, so the function will behave like $1/(x^{1/3})$ as $x\to0$. We can see this in the graph, when the values of $x$ are getting nearer to $0$ from the left, the graph approaches $-\infty$, while when they get nearer to $0$ from the right, the graph approaches $\infty$, just like the graph of the function $y = 1/(x^{1/3})$ as $x\to0$ (the green line).
- As $x\to\pm\infty$, $1/(x^{1/3})$ approaches $0$ while $x^{2/3}$ approaches $\infty$ (x^{2/3}\ge0 for all $x$), so the function will behave like $x^{2/3}$ as $x\to\pm\infty$. We can see this in the graph, when $x\to\pm\infty$, the graph approaches $\infty$, just like the graph of the function $y = 1/(x^{1/3})$ as $x\to\pm\infty$ (the blue line).
