8. The volume of a sphere is determined by the formula $V=\frac{4}{3} \pi R^{3}$. Indicate the independent and dependent variables.
Solution. Here $\frac{4}{3} \pi$ is a constant value. By assigning different values to the radius $R$, we can find the volume of the sphere for each of the given radius values. Thus, the radius $R$ is the independent variable, and the volume of the sphere $V$ is the dependent variable.

The independent variable, i.e., the quantity for which we can assign arbitrary, interesting values, is called the argument. The variable quantity whose values depend on the argument is called the function.

Thus, in Example 7, the variable $t$ is the argument, and $s$ is the function. In Example 8, the variable $R$ is the argument, and $V$ is the function.

Definition 3. A variable quantity $y$ is called a function of a variable quantity $x$ if each value of $x$, taken from its range of variation, corresponds to a unique value of $y$ according to a defined rule.

To show that $y$ is a function of $x$, symbolic notations are used: $y=f(x), y=\varphi(x), y=F(x)$, etc.

Such symbolic notation does not reveal the rule of dependence of $y$ on $x$, but only establishes the fact of the existence of the dependence.

For example, the speed of a freely falling body is a function of time $t$, i.e., $v=f(t)$, and the rule for establishing the correspondence between $t$ and $v$ is known: $v=g t$.

The surface area of a sphere $S$ is a function of its radius $R$, i.e., $S=\varphi(R)$, and the rule for the correspondence between $S$ and $R$ is $S=4 \pi R^{2}$.

Remark. As can be seen from the examples considered above, the argument and the function can be denoted not only by the letters $x$ and $y$, but also by other letters.

The particular value of the function $y=f(x)$ at a given particular value of the argument $x=a$ is symbolically denoted as $f(a)$ or $\left.y\right|_{x=a}$.