Examples

$f (x) = x^{2} + 3 x$

Step 1

Write $f (x) = x^{2} + 3 x$ as an equation.

$y = x^{2} + 3 x$

Step 2

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 $y = x^{2} + 3 x$ 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

Step 3

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[- \frac{9}{4}, \infty)$

Set-Builder Notation:

${y | y \geq - \frac{9}{4}}$

Step 4

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

Not Surjective

Step 5

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