One-to-One Functions — Cheat sheet

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Key method

f(a)=f(b)\Longrightarrow a=b.

Strictly increasing or strictly decreasing functions on an interval are injective there. Restricting the domain can make a noninjective rule injective.

Example

For $f(x)=5x+1$ , equality $5a+1=5b+1$ forces $a=b$ . For $g(x)=x^2$ on $\mathbb R$ , $g(-2)=g(2)$ , so it is not injective. On $[0,\infty)$ , $a^2=b^2$ and nonnegativity force $a=b$ , making the restriction injective.

Avoid this mistake

A graph passing the vertical-line test can still fail the horizontal-line test. It is a function, but may not have an inverse function on the full domain.