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**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*=3*x*+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
**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*=3*x*+2, and we want to ask, “what is the value of *y*=3*x*+2 as *x*=3?” then we want to ask “what is the value of *y*=3*x*+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*=3*x*+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*)=3*x*+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*)=3*x*+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*)=*x*^{2}+3*x*. Now we have that

*f* (2) = 2^{2}+3·2 = 4+6 = 10

*f* (5) = 5^{2}+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) = –2^{2}+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 –2^{2} 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*) = *a*^{2}+3*a*

*f* (2*a*) = (2*a*)^{2}+3·(2*a*) =4*a*^{2}+6*a*

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* (2*a*) = 2*a*^{2}+3·(2*a*) =2*a*^{2}+6*a *: WRONG

More examples:

*f* (*a+*5) = (*a+*5)^{2}+3(*a+*5) = *a*^{2}+10*a*+25+3*a*+15 = *a*^{2}+13*a*+40

*f* (*x+*5) = (*x+*5)^{2}+3(*x+*5) = *x*^{2}+10*x*+25+3*x*+15 = *x*^{2}+13*x*+40

In these cases students often forget the ” whole *x*” thing*. *

WRONG: *f* (*a+5*) = *a+*5^{2}+3*a+*5 = *a*+25+3*a*+5 = 4*a*+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.

Up a level : IB Mathematics

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