Function Notation/Inputs and Outputs — Onlayn dərslik

Let’s look at an alternative way of writing equations that uses a different notation to represent the total amount. We call equations written in this new form functions. With functions, an input value is plugged into the function, some operations are performed on the input value, and then an output value is produced. We can represent the relationships we have talked about in this chapter using function notation, so let’s write an equation as a function to see what functions look like.

For example, if 30 cubic feet of water is being pumped into a pool every minute, we could use

v

to represent the volume of water and

t

to represent the amount of time the pool is being filled. We could say that the volume of water in the pool is a function of time,

t

, because the volume of water is dependent only on the amount of time that has passed. Instead of just using

v

to represent the volume, we can use the notation

v(t)

(pronounced “

v

t

”) to show that

v

is a function of time. The equation representing this relationship could be written as

v(t) = 30t

(the volume of water in the pool is 30 gallons per minute times the number of minutes the water flows into the pool).

The

t

in the parentheses is a placeholder for an input value just as variables are always placeholders. If we wanted to know how much water there would be after 3 minutes, we would use 3 for the value of the function’s input variable

t

. Using function notation, we may be asked to find the value of

v(3)

(pronounced “

v

of 3”), which means the value of

v

when

t

is 3. Whatever value replaces the variable in the parentheses gets substituted in for the variable everywhere it appears in the expression that defines the function, so in this case we replace each instance of

t

with 3.

\begin{aligned} v(t) &= 30t \\ v(3) &= 30(3) \\ v(3) &= 90 \end{aligned}

Realize that the function notation $v(3)$ does not means “

v

times 3.”} Many students get this confused because this is the first time they have seen parentheses used for a purpose other than multiplication. The test will let you know that

v

is a function by using the word “function” in the question. Even if the word “function” is not explicitly stated, it is more likely that

v(3)

is function notation because

v

times 3 (or 3 times

v

) would normally be written as

3v

.

If a function called

f(x)

is defined by the equation

f(x) = 3x + 2

, we can find

f(4)

, the output value when the input value is 4, by substituting 4 for

x

regardless of whether the function represents some real-world situation or it’s a purely abstract mathematical construct; we simply substitute the input value and evaluate the expression to get the output value.