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Functions

For several centuries, mathematicians used equations to represent relationships between variables. For example, in the equation $x + 2y = 4$, as soon as we know a value of

Not all equations provide a choice of which variable can be taken as independent when representing a functional relationship between them. If you want $x + y^2 = 4$, where

If, however, we restrict values of

Today, we distinguish between two ways of representing a relationship between variables. We represent a relationship

We represent a relationship

The equation $x + y^2 = 4$, with

Whether we represent a relationship between variables explicitly or implicitly,

Recall that the graph of a mathematical statement involving two variables is the set of ordered pairs whose coordinates make the statement true. Figure 3.10.1 displays the graph of $x^2 ‑ xy + y^2 = 1$ in rectangular coordinates. That is, Figure 3.10.1 displays the set of points {(

It is clear from the displayed graph of $x^2 ‑ xy + y^2 = 1$ that values of

You can show symbolically that

What restrictions could we put on values of

An examination of Figure 3.10.1 suggests that, with a suitable restriction on values of

If we think of

We are looking for values of

$$(-y)^2 ‑ 4(y^2 ‑ 1) = 0,$$

because when the determinant in Figure 3.10.3 is 0, each of those values of

$$(‑y)^2 ‑ 4(y^2 ‑ 1) = 0$$

gives us $y=\pm \sqrt{\frac{4}{3}}$. Substituting $\sqrt{\frac{4}{3}}$ for

$$x^2 ‑ xy + y^2 = 1$$

gives $x=\pm0.5 \sqrt{\frac {4}{3}}$. Similarly, solving for

$$(‑x)^2 ‑ 4(x^2 ‑ 1) = 0$$

gives $x=\pm \sqrt{\frac{4}{3}}$ and $y=\pm0.5 \sqrt{\frac {4}{3}}$.

By examining Figure 3.10.1 we can tell which values of

$$-0.5 \sqrt{\frac {4}{3}} \le y \le \sqrt{\frac {4}{3}}$$

and

$$-\sqrt{\frac {4}{3}} \le x \le 0.5\sqrt{\frac {4}{3}}.$$

Therefore,

$$x^2 ‑ xy + y^2 = 1, -0.5 \sqrt{\frac {4}{3}} \le y \le \sqrt{\frac {4}{3}},

-\sqrt{\frac {4}{3}} \le x \le 0.5\sqrt{\frac {4}{3}}$$

represents

But we could include more of the ellipse than the part highlighted in red and still have

Even though we have two sets of restrictions on

- Use GC to display a graph of $3x^2 ‑ 2y^2 = 4$. Restrict the domain of
*x*and the domain of*y*in two different ways so that the displayed graph shows*x*as a function of*y*over the largest possible domains of*x*and*y*. You might have to do this in two parts. - Use GC to display a graph of $3x^2 ‑ 2y^2 = 4$. Restrict the domain of
*x*and the domain of*y*in four different ways so that the displayed graph shows*y*as a function of*x*over the largest possible domains of*x*and*y*. You might have to do this in two parts. - Express
*y*as a function of*x*explicitly for each of the following equations. Include restrictions on the domains of*x*and*y*so that they are as large as possible. Use GC to check your answers. - One summer Raul visited a lake on several different evenings and took data regarding temperature, frogs and crickets. At the end of the summer, he organized his data into the following table.
- Make two scatter plots, one for number of chirps per minute and temperature, and one for number of ribbits per minute and temperature. Make temperature the dependent variable (vertical axes) for each.
- Is temperature a function of the number of chirps per minute according to the entries in this table? Explain why or why not. Is temperature a function of the number of ribbits per minute according to the entries in table? Explain why or why not.
- This temperature example gives a clue as to why the term 'function' was chosen as the name for mathematical functions. How is the word 'function' defined outside of mathematics? Use this temperature context to describe how the definition of 'function' relates closely to the idea of a mathematical function. Talk about both chirps and ribbits in your answer.

a) $3x^2-2y^2=4$

b) $x^3-y^3=3$

c) $3x-2y=7$

b) $x^3-y^3=3$

c) $3x-2y=7$

Cricket Chirps per minute | Frog Ribbits per minute | Temperature in deg F |

92 |
4 |
56 |

94 |
3 |
69 |

93 |
5 |
61 |

96 |
2 |
77 |

95 |
4 |
72 |

99 |
3 |
91 |

98 |
4 |
86 |

97 |
5 |
84 |

100 |
5 |
96 |