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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?

Answer

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?

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Answer

The second and the fourth. Can you understand why?

3. Which of these are not graphs of functions?

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Answer

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?

Answer

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.

Answer

Only number 3 above is not a valid function. For this formula, at x = 3, there are two values for y. That's a no-no for a function.

Formula 1 is discontinuous at x = 1, and formula 2 at 2. Formula 4 describes a continuous function. Only its slope is discontinuous at x = 3.

Before continuing to the section on limits, you should look at straight lines.

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