Domain and Range

• From a graph: domain is the "shadow" the graph casts on the $x$-axis (scan left to right); range is the shadow it casts on the $y$-axis (scan bottom to top).
• From a table: domain is the list of $x$-values given; range is the list of $y$-values given (drop duplicates).
• From an equation: ask "what $x$-values would break this rule?" — those are excluded from the domain; everything else is included.

When you read a graph, the endpoints carry the detail. A closed (solid) dot means that point belongs to the graph, so its coordinate is included (use $\le$ or a square bracket). An open (hollow) dot means the graph comes arbitrarily close but that exact point is missing (use $<$ or a round bracket). An arrowhead means the graph keeps going forever in that direction, so the set is unbounded there. The range is often decided by a turning point rather than an endpoint: a parabola opening up with vertex $(2,-5)$ reaches down to $-5$ and no lower, so its range is $\{y\in \mathbb{R}\mid y\ge -5\}$ no matter how far left and right the arms extend. Also notice the difference between a discrete domain (separate values, like the five $x$-values in a table, or whole numbers of tickets) and a continuous domain (every real number in an interval, like time or length).

We describe domain/range with set-builder notation, $\{x\in \mathbb{R}\mid x\ge 3\}$ — read "the set of $x$ in the real numbers such that $x$ is greater than or equal to $3$" — or with interval notation, $[3,\infty )$, where a square bracket $[]$ means the endpoint is included and a round bracket $()$ means it is excluded. Infinity always gets a round bracket, since you can never actually reach it. When a set has a gap, join the pieces with the union symbol $\cup$. For a finite list of values (a discrete domain) use braces with the values listed: $\{-2,0,1,5\}$.

Finding the Domain and Range From a Graph (Interval Notation)

Two algebraic features create automatic domain restrictions. A square root (or any even root) requires its radicand (the expression underneath) to be $\ge 0$, since no real number squared gives a negative, so square roots of negative numbers aren't real. A denominator can never equal $0$, because division by zero has no answer: $\frac{6}{0}$ would have to be a number that gives $6$ when multiplied by $0$, and none exists. So the method for any equation is: set every radicand $\ge 0$, set every denominator $\ne 0$, solve, and combine. If neither feature appears (as in any polynomial, such as $y=3x^2-7x+1$), the domain is all real numbers.

Exam questions often combine both features, and then every condition must hold at once. For $k(x)=\frac{\sqrt{x+2}}{x-3}$, the radicand needs $x+2\ge 0$, so $x\ge -2$, and the denominator needs $x\ne 3$. The domain is the overlap: $\{x\in \mathbb{R}\mid x\ge -2,x\ne 3\}$, or $[-2,3)\cup (3,\infty )$. A square root sitting in the denominator is stricter still: for $\frac{1}{\sqrt{x-1}}$ the radicand must be non-negative and the root must not be zero, so the condition becomes $x-1>0$, giving $x>1$ with a strict inequality.

Domain and range of a function given a formula | Algebra II | Khan Academy

• Quadratic $y=a{(x-h)}^2+k$: the vertex decides it. If $a>0$ the range is $y\ge k$; if $a<0$ it is $y\le k$. Get to vertex form (or use $x=-\frac{b}{2a}$) first.
• Square root $y=a\sqrt{x-h}+k$: the graph starts at $(h,k)$, so the range is $y\ge k$ if $a>0$ and $y\le k$ if $a<0$.
• Reciprocal $y=\frac{a}{x-h}+k$: every $y$ except the horizontal asymptote, $y\ne k$.
• Linear (non-horizontal) with no context: range is all real numbers.
• Real-world model: the situation adds limits the algebra doesn't. Time starts at $0$, a ball's height model stops at the landing time, and a number of people must be a whole number. State these as part of the domain.