# Finite Math 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

Complete the square on the right side of the equation.

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

Consider the vertex form of a parabola.

Find the value of using the formula .

Multiply by .

Reduce the expression by cancelling the common factors.

Factor out of .

Cancel the common factor.

Rewrite the expression.

Divide by .

Find the value of using the formula .

Simplify each term.

Multiply by .

Simplify the numerator.

Remove parentheses around .

Raising to any positive power yields .

Simplify the denominator.

Remove parentheses.

Multiply by .

Multiply by .

Divide by .

Multiply by .

Add and .

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

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.

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

Not Surjective