Next page : Derivatives
Functions
A function is something that for each possible input (x-value) can have one and only one output (y-value). The “something” might be described in many ways:
- By a mathematical expression: y=3x+2 where x is a real number.This in indeed a description of a function since 3 times any real number, correctly calculated, gives one and only one answer.
- By a graph:
(Two marks to the one that can figure out an expression for the above function.)
- By a table:
x y 1 1 2 4 3 7 4 3 5 2 6 1 Each x gives one and only one y, but that does not necessary mean that reach y correspond to one and only one x. We have for example that y=1 correspond to both x=1 and x=6. The important thing is that each allowed x gives one and only one y.
- By a description: Take a natural number and reverse the order of digits in its decimal representation.
This function would turn the number 4711 to 1174.
We can see a function as a kind of recipe for how to turn a number (or something else) to another number (or something else) in such a way that a given allowed input will always give the same output.
The f(x) notation
Suppose we have the function y=3x+2, and we want to ask, “what is the value of y=3x+2 as x=3?” then we want to ask “what is the value of y=3x+2 as x=5?” and so on. Could this not be done in a more efficient way? One way is to give the function y=3x+2 a name. Let us call it f. We could now ask “”what is the value of f as x=5?
We could do this in an even more efficient way by including the value of x, as this “What is f(5)?”. This is read as “What is f of 5?” and it means “what is the value of f as x=5?
We usually define a function like that as
f (x)=3x+2
This is read “f of x is 3x plus 2″ and it means that to find the value of the function f when x has some value we should multiply that value by three and then add two.
The name f is by itself not important, and neither is the variable name. The functions
f (x)=3x+2
g(y)=3y+2
h(z)=3z+2
does all do the same thing. The only important thing with the name is that it allows us to talk about a particular function. We may for example say that the functions f, g and h above are equal because they do the same thing.
The variable name is usually even less important. It is basically just a place holder. Whatever you place instead of the variable name is replacing all occurrences of the same name in the definition.
Some examples
Let us for example redefine f (x) as f (x)=x2+3x. Now we have that
f (2) = 22+3·2 = 4+6 = 10
f (5) = 52+3·5=25+15 = 40
f (–2) = (–2)2+3·(–2) = 4 – 6 = –2
The later case often causes problems. Students often calculate this as
WRONG: f (–2) = –22+3·(–2) = –4 – 6 = –10: WRONG
One has to remember that the definition is that f of x is x squared plus 3 times x, and x squared is x times x. So is x= –2 then we are supposed to square x, the whole x and nothing but the x.
Remember that –22 is calculated as the square of two, then the minus of that, but we want to square the number –2, i.e. we want the result (–2)2=(–2)·(–2)=4.
Let’s look at some slightly more complicated cases.
f (a) = a2+3a
f (2a) = (2a)2+3·(2a) =4a2+6a
Here we have replaced x by an expression. The most common mistake here is to forget that we should square (I repeat) the whole x.
WRONG: f (2a) = 2a2+3·(2a) =2a2+6a : WRONG
More examples:
f (a+5) = (a+5)2+3(a+5) = a2+10a+25+3a+15 = a2+13a+40
f (x+5) = (x+5)2+3(x+5) = x2+10x+25+3x+15 = x2+13x+40
In these cases students often forget the ” whole x” thing.
WRONG: f (a+5) = a+52+3a+5 = a+25+3a+5 = 4a+30 : WRONG
For the example with x the use of x is sometimes confusing. But remember that the x in f (x) is just a placeholder. It has nothing to do with the x in the x+5:part.
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