58. Indicate the intervals of increase of the function

| $x$ | -2 | -1 | 0 | 1 | 2 |
| :--- | ---: | ---: | :--- | :--- | :--- |
| $y$ | 9 | 2 | 0 | 2 | 9 |
Solution. In this case, although the function is given by a table, for clarity, we construct a graph (Fig. 68). It is evident that the function is increasing on the interval $(0, \infty)$.

The most convenient method for defining a function is the third one - the analytical method.

In the analytical method, the dependence between the argument $x$ and the function $y$ is given in the form of a mathematical formula or equation. This formula specifies the operations that need to be performed on the value of the argument to obtain the corresponding value of the function. By assigning different values to the argument $x$, we can calculate the corresponding value of $y$ with the necessary accuracy.

An example of a function defined analytically is the function $y=\frac{2 x^{3}-5}{x+1}$.

The only drawback of the analytical method is the lack of visual clarity. In mathematics, preference is given to this method. Knowing the correspondence rule $y=f(x)$, it is always possible to create a table and plot a graph. Other methods of defining functions do not possess such versatility. In practice, when studying various dependencies, the most convenient approach is to combine different methods of defining functions.

![](https://cdn.mathpix.com/cropped/2024_05_22_db4d450c77c65a914ec1g-162.jpg?height=333&width=354&top_left_y=170&top_left_x=167)

Fig. 68

![](https://cdn.mathpix.com/cropped/2024_05_22_db4d450c77c65a914ec1g-162.jpg?height=304&width=386&top_left_y=200&top_left_x=676)

Fig. 69