DEFINITION: A function can be defined in a variety of ways. Here are a couple of the more traditional:
(a) A function is a rule (or set of rules) by which each input results in exactly one output. The set of all acceptable inputs in called the domain. The set of all resulting outputs is called the range.
(b) A function is a set of ordered pairs in which each first component is matched up with exactly one second component.
What both of these definitions have in common is that each input to the function can result in EXACTLY one output. The usefulness of this idea is that the function has predictable, deterministic outcomes. When things in our world have a relationship in which every input has exactly one output then we describe them as functions.
EXAMPLES OF FUNCTIONS:
(a) The amount of your take home pay.
(b) The amount of your electric bill
(c) Mixing paint to match a particular color
(d) Making cookies from a recipe
(e) The amount of weight an elevator can lift
Because the concept of a function is used so often, a shorthand has been developed...we call it function notation.
Example 1: Identify the parts of the function g(x) = 3x + 7.
(1) the name of the function is g
(2) the letter, x, stands for the input.
(3) the output (based on an input of x) is referred to as g(x)
(4) what the function does (its process) is to take the input, multiply it by 3 and then add 7
The rule g(x) = 3x + 7 is said to be a general rule describing a function. It tells us what to do for ANY input. But there are times when we want to determine the output based on a speicific input. For example, to indicate that we want to put the number 5 into the function g and calculate the output we would write the following:
Example 2: If g(x) = 3x + 7, determine g(5)
Solution:
|
| g(x) |
= 3x + 7 |
| g(5) |
= 3(5) + 7 |
|
= 15 + 7 |
|
=22 |
In other words, when we put the number 5 into the function called g, the output is 22.
COMMON NOTATION ERRORS: This is a good time to point out some common mistakes made by algebra students. Functional notation is very specific and if you are not careful you can end up making a statement that you do not intend. Let's again consider the function in Examples 1 & 2.
We stated that g(x) = 3x + 7 and g(5) = 22
However, g(x) = 3(5) + 7 = 22 would NOT be correct, and
g(5) = 3x + 7 = 3(5) + 7 = 22 is NOT correct
Can you see what makes the last two statements incorrect? Let's look at the statement g(x) = 3(5) + 7 = 22
Remember that g(x) means the output for ANY input, x.
And 3(5) + 7 means the output when the input is 5.
We do not intend to say that the output, g(x), will always be 3(5) + 7, but that is what the statement means. We want to say that when the input is 5 then the output,g(5), will be 22. We say that by writing g(5) = 22.
Now some students will argue that they are eventually stating g(5) = 22. But you cannot have a completely correct statement if part of it is incorrect.
MORE FUNCTION NOTATION: In the function g(x) = 3x + 7, the letter x is actually being used as a place holder. To illustrate this point, we can remove the x. This leads to the following:
g( ) = 3 ( ) + 7
We indicate that something is going into the function by putting it into the parentheses.
So, g(x) represents the output of the function when x represents the input.
And g(5) represents the output of the function when 5 is the input,
and g(a + b) represents the output of the function when a + b is the input,
Here are some examples of functions. In each case notice the function has a name, a letter representing the input, and a specific process by which the output is determined.
Example 3: If f(x) = - 3x2 + 5x - 7, determine f(6)
Solution:
|
| f(x) |
= - 3x2 + 5x - 7 |
| f(6) |
= - 3(6)2 + 5(6) - 7 |
|
= -3 (36) + 30 - 7 |
|
= - 108 + 30 - 7 |
|
= - 85 |
Note that is this example,
- the name of the function is f
- the input is represented by the letter x
- the output (when x is the input) is denoted by f(x)
- f( 6) represents the output of the function when 6 is the input
- the process is described by the statement - 3x2 + 5x - 7
.
Example 4: If g(x) = 2x2 - 5x + 1, determine g(a + b)
| g(x) |
= 2x2 - 5x + 1 |
| g(a+b) |
= 2(a+b)2 - 5(a+b) + 1 |
|
= 2 (a2 + 2ab + b2) - 5a - 5b + 1 |
|
= 2 a2 + 4ab + 2b2 - 5a - 5b + 1 |
- the name of the function is g
- the input is represented by the letter x
- the output (when x is the input) is denoted by g(x)
- g(a+b) represents the output when a + b is the input to the function.
See if you can describe the process that this function represents.
Example 5: If g(x) = 2x2 - 5x + 1, determine g(a) + b
Solution:
This one is actually a lot easier than the one in the previous example. Notice the location of the parentheses in g(a) + b. The input to the function g is just a. g(a) + b directs us to replace x with a in the function g, and then add b. So, we have
g(a) = 2a2 - 5a + 1, and
g(a) + b = 2a2 - 5a + 1 + b
Being able to tell the difference between g(a + b) and g(a) + b is just a matter of learning the language of algebra.
© 2004 Jo Steig
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