SOLUTION: If the function f: R → R such that f(x)= x^2 + 1. Is this function surjective? injective (one-to-one)? bijective? Explain.

Algebra ->  Functions -> SOLUTION: If the function f: R → R such that f(x)= x^2 + 1. Is this function surjective? injective (one-to-one)? bijective? Explain.      Log On


   

Question 1131828: If the function f: R → R such that f(x)= x^2 + 1. Is this function surjective? injective (one-to-one)? bijective? Explain.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29+=+x%5E2+%2B+1 is not injective (one-to-one) onto R
Injective means we won't have two or more "x"s pointing to the same "y".
In other words there are two values of x that point to one y.
Function is said to be injective or one-to-one if every element in the range is an image of at most one element from the domain.
An injective function is called a one-to-one function.
Since x is on the right side of the equation, switch the sides so it is on the left side of the equation.
x%5E2%2B1=y
Add -1 to both sides of the equation: x%5E2=y-1
Take the square root of both sides of the equation to eliminate the exponent on the left side.
x=±sqrt%28y-1%29
The complete solution is the result of both the positive and negative portions of the solution.
x=sqrt%28y-1%29,x=-sqrt%28y-1%29
There is more than y value for some x values, which means that y=x^2+1 is not an equation of a function.

f%28x%29+=+x%5E2+%2B+1 is not surjective onto R
A function f%28x%29 (from set A to B) is surjective if and only if for every y in B, there is at least one x in A such that f%28x%29+=+y, in other words f%28x%29 is surjective if and only if +f%28A%29+=+B.
In simple terms: every B has some A.
Surjective (Also Called "Onto")means that every "y" has at least one matching "x" (maybe more than one).
There won't be a "x" left out.
But,
domain is R+ (all real numbers) and
range (codomain) is
{ f%28x%29 element R : f%28x%29+%3E=+1 }
So, in this function all the negative values in the codomain of f%28x%29 do not have any corresponding x values in the domain of f%28x%29