Download Functions A function is a rule that assigns to each input value a

Functions
A function is a rule that assigns to each input value a unique output value. Below we will consider
some rules and see if they define functions or not.
Let the “ice cream rule” I assign to each person a flavour of ice cream that they like. This rule is
not a function, since the same person (input) may like more than one type of ice cream (output). In
particular I(M organ) = chocolate and I(M organ) = vanilla; that is, Morgan likes both chocolate and
vanilla ice cream, so the rule does not have a reliable single output for the input value “Morgan”.
The “favourite ice cream rule” C assigns to each person their favourite flavour of ice cream. This
rule is a function, since each person may only have one favourite flavour (you have to pick just one!).
So, each time one writes C(M organ), they can be sure that this refers to the same type of ice cream.
The function C has domain the set of all people who like ice cream, and range the set of types of ice cream.
The “food” rule F assigns to each fruit a food that can be made from that fruit. This is not a function
since there are different foods that can be made from the same fruit. For example, one can make either
apple pie or applesauce from apples, so F (apple) = applepie or F (apple) = applesauce.
The “juice” rule J discussed in class is indeed a function, since there is a unique type of juice made
from each fruit; we can be certain that J(apple) refers to apple juice.
The annual rate of inflation in Argentina is a function of the year. For any given year, there is a
unique number representing the rate of inflation in Argentina.
This is represented in the chart by the fact that any value in the “Year” column corresponds to only one
number in the “Inflation rate” column.
With numbers, we may encounter situations where two quantities are related by an equation (often
expresed as an equation in two variable). To determine if the rule defines one quantity y as a function
of the other x, there are at least three good techniques:
1) Try to solve for y in terms of x. If this gives us one solution y for each value of x, the rule is a
function; if not, the rule is not a function.
2) If the graphs are available, use the vertical line test to determine whether it is a function: if each
vertical line grosses the graph no more than once, the rule is a function; if some vertical line meets the
graph more than once, the rule is not a function.
3) If you suspect that the rule is not a function, you can show that indeed it is not by finding a particular
value of x that has more than one associated y-value.
Consider the equation y = 2x − 3. This equation expresses y as a recipe in terms of x, so give
any particular value for x, we know exactly what y should be: take x, multiply by 2, and subtract 3. Thus there is exactly one y value for any given x value, so the equation does express y
as a function of x. Alternatively, consider the graph (plotted by the graphing tool on the website
http://www.flashandmath.com/mathlets/calc/implicit/implicit.html)
and notice that any verical line you can draw will intersect the line at only one point.
Now consder the equation x2 + 4y 2 = 36. We will first check if this expresses y as a function of x by
solving for y.
x2 + 4y 2 = 36
4y 2 = 36 − x2
y 2 = 41 (36 − x2 )
y 2 = 9q
− 41 x2
y = ± 9 − 14 x2
q
q
Then, for a given x value, there are two corresponding y values y = 9 − 14 x2 and y = − 9 − 14 x2 . This
means that we do not have y as a function of x. We can also consider the graph
and notice that there are vertical lines (for example x = 0) that intersect the graph in two points. The
graph fails the vertical line test, so we see again that we do not have y as a function of x. A third way of
seeing that this is not a function is by choosing a particular x value corresponding to multiple y values.
For example, x = 0 corresponds to both y = 3 and y = −3 (we can discover this byplugging x = 0 into
the equation and solving for y, or by noticing the two points (0,3) and (0,−3) on the graph).
Let’s look at the equation x2 + 4y = 16. To see if it describes y as a function of x, solve for y:
x + 4y = 16
4y = 16 − x2
y = 14 (16 − x2 )
y = 4 − 14 x2
We are able to give a recipe for a single y value in terms of x, so this equation does express y as a
function of x. The graph also satisfies the vertical line test:
2
The equation 2|y| − 2x = 4 does not define y as a function of x. Let’s try to solve for y:
2|y| − 2x = 4
2|y| = 2x + 4
|y| = x + 2
So y is a number whose distance from 0 is x + 2. If x + 2 < 0 there are no such numbers, if x + 2 = 0
then y = 0, and if x + 2 > 0 then y = x + 2 or y = −(x + 2) (x + 2 to the right on the number line or
x + 2 to the left). So for all values of x with x + 2 > 0 (x > −2) there are two corresponding y values,
which means that this is not a function. Notice also that the graph does not pass the vertical line test:
Some equations are difficult or even impossible to solve for y. In these cases, the vertical line test is
the our best method for determining which rules define y as a function of x. If I do my job right, we
won’t encounter these equations in any quiz or test problems. The graphs below with equations written
in green define functions, while the graphs with red do not define functions. We can see this by applying
the vertical line test. The top two equations are very difficult to solve for y, and the bottom two are
actually impossible to solve for y!