A function assigns exactly one output in its codomain to every input in its domain.

What are domain and codomain?

The domain is the set of permitted inputs. The codomain is the specified set in which the outputs lie.

What does $f(x)$ mean?

It denotes the value or output of the function $f$ at the input $x$.

What is the image of a function?

It is the set of outputs actually attained: $f(X)=\{f(x)\mid x\in X\}$. It is contained in the codomain.

What is the graph of a function?

The graph is the set of ordered pairs $(x,f(x))$ for all $x$ in the domain.

Discrete mathematicsLesson 2 of 3 in the course Set theory and relations

Functions: Domains, Representations, Composition, and Inverses

5 flashcards · 5 quiz questions · 5 match pairs · 4 fill-in sentences · English

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Summary

A function is a rule that assigns an output to every permitted input. Functions are central to mathematics because they describe relationships, such as how a planet’s position depends on time.

A function from a set $X$ to a set $Y$ assigns exactly one element of $Y$ to each element of $X$ . The set $X$ is the domain; it contains the permitted inputs. The set $Y$ is the codomain; it is the set in which outputs are specified.

All flashcards

What is a function?: A function assigns exactly one output in its codomain to every input in its domain.
What are domain and codomain?: The domain is the set of permitted inputs. The codomain is the specified set in which the outputs lie.
What does $f(x)$ mean?: It denotes the value or output of the function $f$ at the input $x$ .
What is the image of a function?: It is the set of outputs actually attained: $f(X)=\{f(x)\mid x\in X\}$ . It is contained in the codomain.
What is the graph of a function?: The graph is the set of ordered pairs $(x,f(x))$ for all $x$ in the domain.

Quiz questions

For a relation of ordered pairs to define a total function from $X$ to $Y$ , what conditions must it satisfy?Answer: Every input in $X$ has an output in $Y$ , and that output is unique.
For real inputs, what is the domain of $x\mapsto\sqrt{1-x^2}$ ?Answer: $[-1,1]$
If $f:X\to Y$ and $g:Y\to Z$ , how is $(g\circ f)(x)$ evaluated?Answer: Evaluate $f(x)$ first, then apply $g$ to that result.
For a function $f:X\to Y$ , what does $f^{-1}(B)$ denote when $B\subseteq Y$ ?Answer: The inputs in $X$ whose outputs under $f$ belong to $B$ .
When does a function $f:X\to Y$ have an inverse function from $Y$ to $X$ ?Answer: When it is both injective and surjective.

Based on the Wikipedia article Function (mathematics), licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.

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