# Domain, Codomain, Range, Image and Preimage

Can someone please explain to me in layman what each means? Perhaps with some examples with functions (inputs to outputs/numeric values in them)? Especially range, image, and preimage. So far this is my understanding:

• Domain is basically the input $$x$$ in $$f(x)$$.
• Codomain is what $$f(x)$$ produces as an output such as $$y$$ when $$f(x) = y$$.
• Range sounds like codomain but with some restriction?
• Image I have little understanding of but I think it is basically a relation between domain to codomain given that we take a subset of our function (Eg; it is the input to output process of our function given we put a restriction on the domain as $$x$$ can only go from $$0$$ to $$1$$).
• Preimage is just walking backward on the "image" process? (Inverse image?) Going from our "subsetted" output back to our "subsetted" input?

I am very frustrated that I can't seem to grasp these basic concepts so I would greatly appreciate any help from anyone who can break this down for me and help me understand it without too much mathematical notation. Thank you!

• Jun 30, 2019 at 12:06
• Jun 30, 2019 at 12:06
• Jun 30, 2019 at 12:07
• There you can find the definitions: mathematics needs definitions. Jun 30, 2019 at 12:15
• A function $f$ from natural numbers to natural numbers like $x^2$ has as Domain (i.e. the set of "input values") the set $\mathbb N$ and has as Codomain again $\mathbb N$, because all the "output values" are inside $\mathbb N$. Jun 30, 2019 at 12:16

Consider a function for example $$f:R\to R$$ defined by $$f(x)=x^2$$. The domain is the largest possible set of inputs which in this case the set of all real numbers. The codomain is given as $$R$$, the set of all real numbers. The range is the set of all possible outputs which is the interval $$[0,\infty)$$.
The image of a subset $$A$$ of of real numbers is $$f(A)$$ which is the set of all $$f(x)$$ where $$x\in A$$. For example, $$f((-1,1))=[0,1)$$.
The pre-image of a subset $$B$$ of the range is the set $$f^{-1}(B)$$ of all inputs $$x$$ such that $$f(x)$$ is in $$B$$. For example $$f^{-1}([1,4])=[-2,-1]\cup [1,2]$$.
• @Q_A_B_70 Remember that $(-1, 1) = \{x \in \mathbb{R} \mid -1 < x < 1\}$. The square of a number with absolute value less than $1$ must be at least $0$ and less than $1$. Since $f$ is continuous, $f(0) = 0$, and $f(1) = 1$, the function assumes every value between $0$ and $1$, including $0$ but not including $1$ on the interval $[0, 1)$. Jun 30, 2019 at 13:12
• I think it should be $f(\,(-1,1)\,)$. The outer pair of parentheses being the function application, the inner pair being the open interval. Jun 30, 2019 at 13:29