is there any other way to just check, not prove, if a function is injective without this horizontal line test? I could not find any other ways online...
checking if a function is injective without the so called 'horizontal line'
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$\begingroup$
functions
discrete-mathematics
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0It really, really depends on the kind of function you want to test for injectivity and what you know about it. Do you have a special instance in mind? – 2017-02-15
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0What is this new would-be-pedagogical invention "the parallel test" ? This is not mathematical at all ... – 2017-02-15
2 Answers
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A function $f:[a,b]\rightarrow\mathbb{R}$ is injective (on $[a,b]$) iff $f$ is continuous on $[a,b]$ and $f$ is (strictly) monotone on $[a,b]$.
Also, a function $f:[a,b]\rightarrow I\subseteq\mathbb{R}$ is surjective iff $Im(f)=I$ (the image matches the codomain).
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0Your characterization of an injective function is obviously false. – 2017-02-15
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0To illustrate the remark of @NeedForhelp consider $f$ defined on $[0,2]$ by $f(x)=\begin{cases}x & \ \text{ for}0
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It's when we see that the graph of the function only go up or down. Because if not, then we'll easily find 2 distint point with the same y-corrdinate. Also it must be continuous in R.
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0**If** a real-valued function $f$ is **continuous** and injective then it's true that it is monotonic. However if $f$ is not continuous that's not the case in general. – 2017-02-15
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0@Lê Đức Minh As you are new on this site, it is maybe difficult for you to situate the level, but surely it is above "familiar sentences" that you could possibly use with a young student of 15-16 years who begins with curves and has difficulties with formal mathematics. – 2017-02-15