# Algebra Examples

Function is said to be a surjection or onto if every element in the range is an image of at least one element of the domain. This means the range of must be all real numbers for the function to be surjective. If the range is not all real numbers, it means that there are elements in the range which are not images for any element from the domain.

Range should be all real numbers

Rewrite the equation in vertex form.

Complete the square for .

Use the form , to find the values of , , and .

Consider the vertex form of a parabola.

Substitute the values of and into the formula .

Multiply by .

Find the value of using the formula .

Simplify each term.

Raise to the power of .

Multiply by .

Subtract from .

Substitute the values of , , and into the vertex form .

Set equal to the new right side.

Use the vertex form, , to determine the values of , , and .

Since the value of is positive, the parabola opens up.

Opens Up

Find the vertex .

The range of a parabola that opens up starts at its vertex and extends to infinity.

Interval Notation:

Set-Builder Notation:

The range is not all real numbers, which means there is that is an image for no element from the domain.

Not Surjective