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Section 3.10 Functions


The concept of function permeates modern mathematics. The fundamental idea of a function is that values of variables are related in such a way that knowing the value of one variable tells you exactly one value of the other. Though today’s formal definition of a function evolved over the past several centuries, the basic idea never changed — a function is a relationship between variables that has the property that any value of one variable determines exactly one value of the other.

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 y, the value of x is determined uniquely (there is only one value of x that is determined by a given value of y). Similarly, as soon as you know a value of x in the equation $x + 2y = 4$, the value of y is determined uniquely. If you consider x the independent variable in $x + 2y = 4$, then y is a function of x. If you consider y the independent variable in $x + 2y = 4$, then x is a function of y. Thus, it is important that anyone who writes an equation to represent a functional relationship between two variables also state which variable is independent when either could be taken as independent.

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 x and y are real numbers, to represent a functional relationship, then you have no choice about which variable to take as independent. If you take x to be independent, then when x = 0, y = 2 or y = -2. This means that there would be some value of x for which y is not determined uniquely. On the other hand, no matter the value we assign to y, the value of x is determined uniquely. Therefore in the equation $x + y^2 = 4$, x is a function of y but y is not a function of x.

If, however, we restrict values of y in  $x + y^2 = 4$ to y ≥ 0, then any value of x less than or equal to 4 would determine a value of y uniquely. Therefore, in  $x + y^2 = 4$, where x ≤ 4 and y ≥ 0, either x or y could be taken as independent and  $x + y^2 = 4$ would represent a functional relationship between x and y.

Today, we distinguish between two ways of representing a relationship between variables. We represent a relationship explicitly by writing the dependent variable to the left of the equal sign and on the right side writing a description of how to determine the dependent variable’s value from a value of the independent variable.

We represent a relationship implicitly by representing the relationship in a way that we must give additional information about which variable to take as independent.

The equation $x + y^2 = 4$, with y the independent variable, represents a relationship between values of x and values of y implicitly. The same relationship between x and y is represented explicitly by the equation $x=4‑y^2$. We do not need to state which variable is independent when we represent a relationship between variables explicitly. It is good practice in both cases to state the domain of values that the independent variable can have. In this book we use the custom that an independent variable’s value can be any real number unless stated otherwise, or unless the context makes the restriction clear.

Whether we represent a relationship between variables explicitly or implicitly, for a relationship between variables to be called a function, all values of the independent variable must determine exactly one value of the dependent variable.

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 {(x, y), where x and y are real numbers such that $x^2 ‑ xy + y^2 = 1$}.

Displayed graph x^2-xy+y^2=1

Figure 3.10.1. The displayed graph of $x^2 ‑ xy + y^2 = 1$.

It is clear from the displayed graph of $x^2 ‑ xy + y^2 = 1$ that values of x are not determined uniquely from values of y when 0 ≤ y ≤ 1. We can show this by displaying the graph of y = 1 on top of the graph of $x^2 ‑ xy + y^2 = 1$, 0 ≤ y ≤ 1 (Figure 3.10.2). Figure 3.10.2 shows that the value y = 1 determines two values of x, namely x = 0 and x = 1. When animated, Figure 3.10.2 also shows that any value of y, 0 ≤ y ≤ 1, determines two values of x.


Figure 3.10.2. Each value of y, 0 ≤ y ≤ 1, determines two values of x.

You can show symbolically that y = 1 determines two values of x in $x^2 ‑ xy + y^2 = 1$. Substitute 1 for y in $x^2 ‑ xy + y^2 = 1$, getting $x^2 ‑ 1x + 12 = 1$. The solutions of the equation $x^2 ‑ x + 1 = 1$ are x = 0 and x = 1. Therefore y = 1 determines two values of x.

What restrictions could we put on values of x so that values of x are determined uniquely by values of y between 0 and 1? Clearly, the restriction $-1 ≤ x ≤ 0$ when $0 ≤ y ≤ 1$ works. Thus, $x^2 ‑ xy + y^2 = 1$ represents x as a function of y when y is taken as the independent variable,  $-1 ≤ x ≤ 0$, and $0 ≤ y ≤ 1$.

An examination of Figure 3.10.1 suggests that, with a suitable restriction on values of x, there are values of y smaller than 0 and values of y larger than 1 for which we can determine x as a function of y. We can determine the appropriate values by finding values of y in $x^2 ‑ xy + y^2 = 1$ that give exactly one value of x, by finding values of x that give exactly one value of y, and then use those values to restrict y and x.

If we think of y as a parameter in $x^2 ‑ xy + y^2 = 1$ instead of as a variable, then we can use the quadratic formula to find values of y that give just one value of x, as shown in Figure 3.10.3.

Quadratic formula

Figure 3.10.3. The quadratic formula applied to $x^2 ‑ xy + y^2 = 1$ by thinking of y as a parameter.

We are looking for values of y that are solutions to
$$(-y)^2 ‑ 4(y^2 ‑ 1) = 0,$$
because when the determinant in Figure 3.10.3 is 0, each of those values of y will determine exactly one value of x. We also want values of x that determine exactly one value of y. Solving for y in
$$(‑y)^2 ‑ 4(y^2 ‑ 1) = 0$$
gives us $y=\pm \sqrt{\frac{4}{3}}$. Substituting $\sqrt{\frac{4}{3}}$ for y in
$$x^2 ‑ xy + y^2 = 1$$
gives $x=\pm0.5 \sqrt{\frac {4}{3}}$. Similarly, solving for x in
$$(‑x)^2 ‑ 4(x^2 ‑ 1) = 0$$
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 x go with which values of y. We want 
$$-0.5 \sqrt{\frac {4}{3}} \le y \le \sqrt{\frac {4}{3}}$$
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 x as a function of y (see the portion of Figure 3.10.4’s graph in red).

But we could include more of the ellipse than the part highlighted in red and still have x as a function of y. We could also include the section of the ellipse in blue (Figure 3.10.4). Figure 3.10.4 shows the restrictions on x and y that generates the blue section of the ellipse.

Highlighted graph that is function

Figure 3.10.4. The highlighted section of the displayed graph of $x^2 ‑ xy + y^2 = 1$ represents x as a function of y.

Even though we have two sets of restrictions on x and y in Figure 3.10.4, the restrictions together are such that every value of y in its domain (namely, $-\sqrt{\frac {4}{3}} \le y \le \sqrt{\frac {4}{3}}$ is related to exactly one value of x. Therefore, with these restrictions on x and y, $x^2 ‑ xy + y^2 = 1$ represents x as a function of y.

Exercise Set 3.10

  1. 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.

  2. 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.

  3. Use GC to display the the graph of $x^2 ‑ xy + y^2 = 1$, as in Figure 3.15. Restrict the domain of x and the domain of y so that the resulting 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.

  4. 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.

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

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

    c) $x^3-xy+y^3=0$

    d) $3x-2y=7$

  6. 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.

  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

    1. 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.

    2. 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.

    3. 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.