6.3 Classes of Functions

There are certain properties of functions that we can use to group functions together into similarly-behaving types. Let $f$ be a function with domain and range in $\mathbb{R}$ .

Definition 6.5.

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$f$ is an even function $\iff f(x)=f(-x)\ \forall x\in\operatorname{dom}(f)$ . This can be visualised as symmetry about the y-axis of the graph.
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$f$ is an odd function $\iff-f(x)=f(-x)\ \forall x\in\operatorname{dom}(f)$ . This can be visualised as $180^{\circ}$ rotational symmetry about the origin of the graph.

It is possible for a function to be neither even nor odd.

Definition 6.6.

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$f$ is an injection (or one-to-one) $\iff$

$\forall x,x^{\prime}\in\operatorname{dom}(f),\ {f(x)=f(x^{\prime})\implies x=x% ^{\prime}}.$

Every element in the range is the image of at most one element from the domain.
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$f$ is a surjection (or onto) $\iff$

$\forall y\in\operatorname{range}(f),\ \exists x\in\operatorname{dom}(f)\ \text% {s.t.}\ y=f(x).$

Every element in the range is reached by at least one element from the domain.
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f is a bijection (or one-to-one correspondence) $\iff f$ is injective and $f$ is surjective. Each element of the range is mapped to by exactly one element from the domain.

It is possible for a function to be neither injective nor surjective.