The function f:[0,∞)→R is defined by f(x)=2x2+3.
Domain: [0,∞) (all non-negative real numbers)
Codomain: R (all real numbers)
To check if the function is one-one, assume f(x1)=f(x2):
2x12+3=2x22+3
2x12=2x22
x12=x22
Since both x1,x2∈[0,∞), both are non-negative:
x1=x2
If outputs are equal, inputs must be equal. The function is one-one.
To check if the function is onto, find the range.
When x=0:
f(0)=2(0)2+3=3
This is the minimum value.
As x increases, f(x) increases since x2 increases.
As x→∞, f(x)→∞
Range of f=[3,∞)
Codomain: R (all real numbers)
Range: [3,∞) (only numbers greater than or equal to 3)
Since Range = Codomain, the function is not onto.
Values like f(x)=0, f(x)=1, or f(x)=−5 cannot be achieved as they are less than 3.
The function is one-one but not onto.