Injective function
An injective function, also called one-to-one, never sends two different inputs to the same output.
Formally, a Function is injective when for all . Equivalently,
eq x_2\Rightarrow f(x_1) eq f(x_2).$$ ## Examples The function $f:\mathbb R\to\mathbb R$, $f(x)=2x+1$, is injective. If $2x_1+1=2x_2+1$, then $x_1=x_2$. The function $g:\mathbb R\to\mathbb R$, $g(x)=x^2$, is not injective because $g(2)=g(-2)=4$. Restricting the domain to $[0,\infty)$ makes the same formula injective. ## Why it matters Injectivity is the “no information lost by merging” condition. If a mapping is not injective, you cannot uniquely recover the input from the output. This is why injectivity is required for an [[Inverse functions|inverse function]] on the original domain. In data or physics language, a non-injective measurement maps multiple possible states to the same reading. Extra information or restrictions are needed to reconstruct the original state. Related: [[Surjective function]], [[Bijective function]], [[Domain]], [[Mapping]]. ## Why it matters Injectivity is the "no information lost" condition. If two different inputs can collapse to the same output, then you cannot uniquely recover the input from the output. Example:f(x)=x
is not injective on $\mathbb{R}$ because $f(1)=f(-1)=1$. But it is injective on $[0,\infty)$, because restricting the [[Domain]] removes the duplicate branch. This is exactly why square-root inverses need domain restrictions.