math
9th Grade - Functions

9th Grade lesson

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 function, and see what happens when you give it the numbers 0, 1, 4, 9.

image calculation

It enables us to plot the graph of the function :
graph function with image and preimage

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 graph equation solve thanks to a graph, you have to find all the x's on the X-axis such that equation. 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.
graph

Domains

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

You can't calculate f(2) because it's impossible to divide by zero :
no way to divide bye 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 function,its domain is domain. Which means all the numbers except 2.
If function, then domain, 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) :
increasing function

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) :
decreasing function


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 definition of even function. The 2 images have the same height.
even function
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 odd inverse function.
odd function


Square function

The square function is square function.

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 :
even function example
parabol

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 domain function inverse. It is odd :
odd function
hyperbol




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9th Grade function

lesson, problems