Images, preimagesAs 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.
DomainsBe 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 functionA 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) :
ParityWell, one day it will be successful...
More seriously :