About One to One Functions:
One to one functions are defined as a relation of x,y where for every x there is only one value of y.
These functions have to pass the horizontal line test, because if there are multiple x values for the same y value, then the relation is not one to one.
The horizontal line test means that if you put a pencil horizontally on the graph of a given relation, and moves it up and down the y-axis, there would be only one point touching the pencil at a time.
Definition of One-to-One :
Example 1:
This is an example of a one to one function. We can find out using the definition of a one to one function!
f (x) = 3x - 4
This is a one to one function because it passes the horizontal line test since it is linear, and makes the above equation true.
Example 2:
Take a look at the graph of this parabola. Shown here is the function f(x)=x^2
These functions have to pass the horizontal line test, because if there are multiple x values for the same y value, then the relation is not one to one.
The horizontal line test means that if you put a pencil horizontally on the graph of a given relation, and moves it up and down the y-axis, there would be only one point touching the pencil at a time.
Definition of One-to-One :
f(a)=f(b), a=b
Example 1:
This is an example of a one to one function. We can find out using the definition of a one to one function!
f (x) = 3x - 4
f(a)=f(b)
3(a)-4=3(b)-4
3a-4=3b-4
+4 +4
3a=3b
a=b
It's a... One to One Function!!
Example 2:
Take a look at the graph of this parabola. Shown here is the function f(x)=x^2
Now, looking at this graph, we can decide it is not a one to one funciton because f(x)=x^2 does not pass the horizontal line test :(
Good to Know!
If a relation is a one-to-one function, it cannot be an even function
The inverse of a one-to-one function is also a function
Good to Know!
If a relation is a one-to-one function, it cannot be an even function
The inverse of a one-to-one function is also a function
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