Functions: injective

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Question statement

Informally, when a function has exactly one input value for every output value we say it is injective or one-to-one.

Formally, a function $f(x)$ is said to be injective when $f(x_1) = f(x_2)$ implies that $x_1=x_2$ .

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			x1	decimal	dec("2.2")
			floorx1	decimal	dec("2")

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Part a shows that to prove a function is not injective, find specific values of $x_1,x_2$ such that $f(x_1) = f(x_2)$ but $x_1 \neq x_2$ .

Part b, this same method can be used to show that

$\sin$ is not injective since $\sin(0) = \sin(\pi)$ but $0 \neq \pi$ .
$g(x)$ is not injective since $g(-1) = g(1)$ but $-1 \neq 1$ .

There are two common methods to show a function is injective.

If the function is continuous and has strictly positive (or strictly negative) derivative, then the function is injective. This is useful to prove that $\tan(x)$ is injective because it is continuous on $(-\pi/2,\pi/2)$ and has derivative $\sec^2(x) > 0$ .
For some simple functions it is possible to prove the injective property directly. For example if you assume

$f(x_1) = f(x_2)$

by definition of $f$

$2x_1 = 2x_2$

and dividing by $2$ :

$x_1=x_2$ .

Hence $f(x) = 2x$ is injective.

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Daniel Mansfield 7 years, 8 months ago

Daniel Mansfield 7 years, 8 months ago

Daniel Mansfield 7 years, 9 months ago

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Functions: injective

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Taxonomy: mathcentre

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Taxonomy: Context

Contributors

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Daniel Mansfield 7 years, 8 months ago

Daniel Mansfield 7 years, 8 months ago

Daniel Mansfield 7 years, 9 months ago

Error in variable testing condition

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Generated value: decimal

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Number entry

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Generated value: `decimal`