Editing Injective function (section)

== Proving that functions are injective ==

A proof that a function <math>f</math> is injective depends on how the function is presented and what properties the function holds.
For functions that are given by some formula there is a basic idea.
We use the definition of injectivity, namely that if <math>f(x) = f(y),</math> then <math>x = y.</math><ref>{{cite web|last=Williams|first=Peter|title=Proving Functions One-to-One|url=http://www.math.csusb.edu/notes/proofs/bpf/node4.html |date=Aug 21, 1996 |website=Department of Mathematics at CSU San Bernardino Reference Notes Page |archive-date= 4 June 2017|archive-url=https://web.archive.org/web/20170604162511/http://www.math.csusb.edu/notes/proofs/bpf/node4.html}}</ref>

Here is an example: 
<math display="block">f(x) = 2 x + 3</math>

Proof: Let <math>f : X \to Y.</math>  Suppose <math>f(x) = f(y).</math>  So <math>2 x + 3 = 2 y + 3</math> implies <math>2 x = 2 y,</math> which implies <math>x = y.</math>  Therefore, it follows from the definition that <math>f</math> is injective.

There are multiple other methods of proving that a function is injective.  For example, in calculus if <math>f</math> is a differentiable function defined on some interval, then it is sufficient to show that the derivative is always positive or always negative on that interval.  In linear algebra, if <math>f</math> is a linear transformation it is sufficient to show that the kernel of <math>f</math> contains only the zero vector.  If <math>f</math> is a function with finite domain it is sufficient to look through the list of images of each domain element and check that no image occurs twice on the list.

A graphical approach for a real-valued function <math>f</math> of a real variable <math>x</math> is the [[horizontal line test]]. If every horizontal line intersects the curve of <math>f(x)</math> in at most one point, then <math>f</math> is injective or one-to-one.