# 3 - Functions

## Images, preimages

As we have seen in the 8th Grade lesson, a function is a machine to which you give numbers and gives you other numbers in return. Let's take the function , and see what happens when you give it the numbers 0, 1, 4, 9.

It enables us to plot the graph of the function :

1 is called the image of 4 under f.
9 is called a preimage of 2 under f.

Don't forget that the x values have to be found on the X-axis (horizontal) and that the f(x) values are to be found on the Y-axis. If you want to solve the equation thanks to a graph, you have to find all the x's on the X-axis such that . You can draw a horizontal line of height 1, and find the values of x where the line and the graph intersect. Solving such an equation is equivalent to finding the preimages of 1 under f.

## Domains

Be careful : you can't always calculate the number the function will give. For example,

You can't calculate f(2) because it's impossible to divide by zero :

The closer to zero the number under the bar, the greater the result. Actually, when you divide by zero, the result it infinite and infinity is not a number, that's why we say that it's impossible to divide by zero.
The set of values of x for which it is possible to calculate f(x) is called the domain of the function. Consider the function ,its domain is . Which means all the numbers except 2.
If , then , which means all the positive numbers (You can't calculate the square root of a negative number).

## Monotony of a function

A function is said to be increasing if its graph goes up. In that case, with two numbers a<b on the X-axis, we get f(a)<f(b) :

A function is said to be decreasing if its graph goes down. In that case, with two numbers a<b on the X-axis, we get f(a)>f(b) :

## Parity

Well, one day it will be successful...
More seriously :

 An even function is a function whose graph is symmetric with respect to the Y-axis (vertical axis). In that case, for every x, we have . The 2 images have the same height.
 An odd function is a function whose graph is symmetric with respect to the origin (the point 0). In that case, for every x, we have .

## Square function

The square function is .

 x -2 -1 0 1 2 f(x) 4 1 0 1 4

Its graph is called a parabola. It is decreasing then increasing. It's even :

## Inverse function

The inverse function is f(x) = 1/x.

 x - 2 - 1 -1/2 1/2 1 2 1/x -1/2 -1 -2 2 1 1/2

Its graph is called an hyperbola. It is always decreasing. Its domain is . It is odd :

>>> Signs charts lesson >>>