Domain and Range

When your group graphed the thrown-ball function at the boards, someone asked the question this page answers: “wait — are we allowed to plug in $t=-3$?” The domain of a function is the set of inputs it accepts; the range is the set of outputs it can produce. Every function drags both sets around with it, and stating them is part of describing the function.

The four parents

Function	Domain	Range
$f(x)=x$	all real numbers	all real numbers
$f(x)=x^2$	all real numbers	$y\ge 0$
$f(x)=\sqrt{x}$	$x\ge 0$	$y\ge 0$
$f(x)=\frac{1}{x}$	$x\ne 0$	$y\ne 0$

Each restriction has a reason you can say out loud: a square is never negative; a square root refuses negative inputs; division by zero has no meaning, and $\frac{1}{x}$ can never actually be zero. Say the reason, not just the rule — the reason survives in your memory long after the rule fades.

Domain and range travel with the graph under transformations. Slide $f(x)=\sqrt{x}$ right 4 to get $g(x)=\sqrt{x-4}$ and the domain slides too: $x\ge 4$. Lift $y=x^2$ by 1 and the range lifts: $y\ge 1$. If you can sketch the image, you can read both sets straight off it — Using Desmos makes the check instant.

When context narrows the view

Algebra is generous; reality is not. The height of a thrown ball, $h(t)=-5t^2+20t$, is algebraically happy with any real $t$ — but the throw starts at $t=0$ and the ball lands at $t=4$, so the model’s domain is $0\le t\le 4$ and its range is $0\le h\le 20$. A cost function for concert tickets only accepts whole numbers of tickets. Whenever a function models something, finish the sentence: “…and the context restricts the domain to —.” Many of our Graph Talks hinge on exactly that finishing move.

Domain and Range Practice runs you through both jobs: reading the sets from equations, and letting a context shrink them.

Curriculum connection

A1.3

explain the meanings of the terms domain and range, through investigation using numeric, graphical, and algebraic representations of the functions $f(x)=x$, $f(x)=x^2$, $f(x)=\sqrt{x}$, and $f(x)=\frac{1}{x}$; describe the domain and range of a function appropriately (e.g., for $y=x^2+1$, the domain is the set of real numbers, and the range is $y\ge 1$); and explain any restrictions on the domain and range in contexts arising from real-world applications Sample problem: A quadratic function represents the relationship between the height of a ball and the time elapsed since the ball was thrown. What physical factors will restrict the domain and range of the quadratic function?

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A1.9

sketch graphs of $y=af(k(x-d))+c$ by applying one or more transformations to the graphs of $f(x)=x$, $f(x)=x^2$, $f(x)=\sqrt{x}$, and $f(x)=\frac{1}{x}$, and state the domain and range of the transformed functions Sample problem: Transform the graph of $f(x)$ to sketch $g(x)$, and state the domain and range of each function, for the following: $f(x)=\sqrt{x}$, $g(x)=\sqrt{x-4}$; $f(x)=\frac{1}{x}$, $g(x)=-\frac{1}{x+1}$.

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