Refresher - Formal Definition

Refresher

The domain of a function is the set of "input" or argument values for which the function is defined. That is, the function provides an "output" or "value" for each member of the domain.

Formal Definition

Given a function ${\displaystyle f:X\to Y}$, the set ${\displaystyle X}$ is the domain of ${\displaystyle f}$; the set ${\displaystyle Y}$ is the codomain of f. In the expression ${\displaystyle f\left(x\right),x}$ is the argument and ${\displaystyle f\left(x\right)}$ is the value. One can think of an argument as an input to the function, and the value as the output.
The image (sometimes called the range) of ${\displaystyle f}$ is the set of all values assumed by ${\displaystyle f}$ for all possible ${\displaystyle x}$; this is the set ${\displaystyle \{f\left(x\right):x\in X\}}$ .The image of ${\displaystyle f}$ can be the same set as the codomain or it can be a proper subset of it. It is in general smaller than the codomain; it is the whole codomain if and only if ${\displaystyle f}$ is a surjective function.
A well-defined function must carry every element of its domain to an element of its codomain.
For example, the function f defined by ${\displaystyle f\left(x\right)=\frac{1}{x}}$ has no value for ${\displaystyle f\left(0\right)}$. Thus, the set of all real numbers, ${\displaystyle ℝ}$ , cannot be its domain. In cases like this, the function is either defined on ${\displaystyle ℝ-\left\{0\right\}}$ or the "gap is plugged" by explicitly defining ${\displaystyle f\left(0\right)}$. If we extend the definition of ${\displaystyle f}$ to
${\displaystyle f\left(x\right)=\frac{1}{x}}$, for ${\displaystyle x\ne 0}$,
${\displaystyle f\left(0\right)=0}$,
then f is defined for all real numbers, and its domain is ${\displaystyle ℝ}$ .
If the expression defining function ${\displaystyle f}$ contains a square root then the expression under the radical has to be greater than or equal to zero. For example, the function ${\displaystyle f\left(x\right)=\sqrt{x}}$ is defined for all ${\displaystyle x\ge 0}$or ${\displaystyle [0,+\infty )}$.
In short: when finding the domain, remember:

• The denominator (bottom) of a fraction cannot be zero
• The values under a square root sign must be positive

The range of a function is the complete set of all possible resulting values of the dependent variable of a function, after we have substituted the values in the domain.