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# Functions and graphs

The volume of a sphere depends on its radius. Let v denote the volume and r the radius.

v is a function of r, written like this: v(r). The values of r go from 0 to ∞. This is the domain of the function and it extends from 0 to ∞.

For each value of r in this domain, the function assigns a number, which is the volume of the sphere. The volume v extends from 0 to ∞. This is the range of the function.

The function is given by the formula

$v = \frac43 \pi r^3$

We also refer to r as the independent variable and v as the dependent variable.

We can plot v vs r:

Volume of a sphere as a function of the radius

A function assigns one number in the range to each number in the domain. It is not a function if more than one number is assigned to a value of the independent variable.

For example, √x is not a function if we mean by √ both the positive and the negative roots. If we restrict ourselves to the positive (or negative) root, then it describes a function.

1. Write down the surface area of a cube as a function of its length. Write down the volume of a cube as a function of its surface area. Can you write the length of a cube as a function of its volume? Of its surface area?

The surface area, s, of a cube as a function of its length, l:

$s = 6 l^2$

The length of the side of a cube is a positive quantity. You don't have a cube with a side of negative length. Therefore, you can write the length of the cube as

$l = \sqrt{\frac{s}{6}}$

as long as you are only referring to the positive root. You can then write

$v = l^3 = {\frac{s}{6}}^{\frac32}$

The cube root of a positive number is positive. Therefore, there is no problem in expressing the length of a cube as a function of its volume:

$l = v^{\frac13}$

2. Which of these are not graphs of functions?

The second and the fourth. Can you understand why?

3. Which of these are not graphs of functions?

The third and the fourth. For these two graphs, you do not have a unique value of y for every value of x.

4. Which of the above graphs(in questions 2 and 3) represent continuous functions and which represent discontinuous functions?

The first and third graphs in question 2 represent continuous functions. The first and second graphs in question 3 represent discontinuous functions.

5. Which of these formulas are valid descriptions of functions? Which of them represent continuous functions?

1. $\begin{array}{rcl}f(x) &= &4,\ 0 \leq x \leq 1 \\ &= &8,\ 1 < x < \infty\end{array}$
2. $\begin{array}{rcl}f(x) &= &2,\ 0 \leq x < 2 \\ &= &6,\ 2 \leq x < \infty\end{array}$
3. $\begin{array}{rcl}f(x) &= &1,\ 0 < x \leq 3 \\ &= &2, 3 \leq x < \infty\end{array}$
4. $\begin{array}{rcl}f(x) &= &x,\ 0 \leq x \leq 3 \\ &= &21 - 6x,\ 3 \leq x \leq 7\end{array}$

Sketch the graphs of all the above formulas.