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Domain and Range

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Domain and Range - Complete Interactive Lesson

Part 1: What Domain and Range Mean

🎯 Domain and Range

Part 1 of 5 — What Domain and Range Mean

Topics in This Part

Section
Inputs, Outputs, and Functions
Defining Domain and Range
Three Ways to Write a Set of Numbers

🔑 Key Concept: Every function is a machine: you feed it an input ( $x$ ) and it returns an output ( $y$ ). The domain is the collection of all legal inputs; the range is the collection of all outputs the machine can produce.

Inputs, Outputs, and Functions

Think of $f(x) = 2x + 1$ as a function machine. Drop a number in for $x$ , and out comes a result:

Input $x$	Rule $2x+1$

Concept Check 🎯

Domain & Range of a Set of Points

When a relation is just a list of points, the domain is the set of all the $x$ -coordinates and the range is the set of all the $y$ -coordinates. List each value once, in increasing order.

Example

$\{(-3,\,4),\;(0,\,1),\;(2,\,4),\;(5,\,-2)\}$

Read the Coordinates 🧮

Use the relation $\{(-1,\,7),\;(3,\,2),\;(4,\,7),\;(6,\,0)\}$ .

Three Ways to Describe a Set

The same set of numbers can be written three ways. You will meet all three:

Notation	"All real numbers from $-2$ up to (not including) $5$ "
Words	$x$ is at least $-2$ and less than $5$

Translate the Notation 🔽

Match each inequality to its correct interval.

Part 2: Reading Domain & Range from Graphs

🎯 Domain and Range

Part 2 of 5 — Reading Domain & Range from Graphs

🔑 The Idea: To find a domain from a graph, "flatten" the curve onto the $x$ -axis and ask how far left and right it reaches. For the range, project onto the $y$ -axis and ask how far down and up it reaches.

Project Onto the Axes

Domain: scan left → right. What is the smallest $x$ the graph touches? The largest? That horizontal sweep is the domain.
Range: scan bottom → top. What is the lowest $y$ ? The highest? That vertical sweep is the range.

Part 3: Domain from an Equation

🎯 Domain and Range

Part 3 of 5 — Domain from an Equation

🔑 The Idea: When you only have a formula, start by assuming every real number is allowed, then remove the values that break a rule. Two things break the rules: dividing by zero and taking the square root of a negative.

The Two Forbidden Moves

Trouble	The rule	What to do
Division by zero	The denominator can never equal $0$ .	Set the denominator $= 0$ , solve, and exclude that $x$ .

Part 4: Range & Real-World Restrictions

🎯 Domain and Range

Part 4 of 5 — Range & Real-World Restrictions

🔑 The Idea: Range answers "what outputs are possible?" For familiar shapes you can reason it out from the vertex or the floor of a square root. And in word problems, context can shrink a domain that the math alone would allow.

Range of Common Functions

Function	Range	Why
Line $y = mx + b$ (with $m \ne 0$ )

Part 5: Mixed Practice & Mastery Check

🎯 Domain and Range

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) define domain and range, (2) read them off a graph, (3) find a domain from an equation, and (4) account for range and real-world limits. Let's put it together.

Quick Reference

Situation	What to do
Set of points	Domain = distinct $x$ 's; Range = distinct $y$ 's
Graph	Domain = left↔right sweep; Range = down↔up sweep
Polynomial	Domain $= (-\infty,\, \infty)$

The domain is the set of every $x$ you are allowed to put in.
The range is the set of every $y$ that can come out.

💡 A handy phrase: "Domain = $x$ , Range = $y$ ." Domain is read left-to-right along the horizontal axis; range is read bottom-to-top along the vertical axis.

Domain: $\{-3,\,0,\,2,\,5\}$ — the $x$ -coordinates.
Range: $\{-2,\,1,\,4\}$ — the $y$ -coordinates, with the repeated $4$ written only once.

⚠️ A value that appears more than once is still listed a single time. Here $4$ shows up twice as a $y$ -coordinate, but the range contains just one $4$ .

1) How many distinct values are in the domain? $\,?$ 2) How many distinct values are in the range? $\,?$ 3) What is the smallest number in the domain? $\,?$

The bracket rules for interval notation:

Square bracket $[\;]$ → endpoint is included (matches $\le$ or $\ge$ ).
Round bracket $(\;)$ → endpoint is excluded (matches $<$ or $>$ ).
$\infty$ and $-\infty$ always get a round bracket — you can never "reach" infinity.

🔑 All real numbers is written $(-\infty,\, \infty)$ . We use this constantly in Part 3.

Example: a parabola opening up

The graph of $y = x^2$ stretches forever left and right, so its domain is all real numbers, $(-\infty,\, \infty)$ .

Its lowest point is the vertex at $y = 0$ , and it climbs forever upward, so its range is $y \ge 0$ , written $[0,\, \infty)$ .

Feature	$y = x^2$
Domain	$(-\infty,\, \infty)$
Lowest output	$0$ (at the vertex)
Range	$[0,\, \infty)$

💡 A plain line like $y = 2x + 1$ also has domain and range equal to $(-\infty,\, \infty)$ — it covers every $x$ and every $y$ .

Concept Check 🎯

Open vs. Closed Endpoints

Graphs often start or stop. The dot at an endpoint tells you whether that point is included:

Closed (filled) dot ● → the point is part of the graph → use a square bracket.
Open (hollow) dot ○ → the point is not part of the graph → use a round bracket.

Example

A line segment is drawn from a closed dot at $(-2,\,1)$ to an open dot at $(3,\,6)$ .

Domain: $x$ runs from $-2$ (included) to $3$ (excluded) → $[-2,\, 3)$ .
Range: as $x$ goes from $-2$ to $3$ , $y$ rises from $1$ (included) to $6$ (excluded) → .

⚠️ Match each bracket to its own dot. A graph can be closed on one end and open on the other, like $[-2,\, 3)$ above.

Pick the Right Bracket 🔽

A curve runs from a closed dot at $(-5,\,2)$ to an open dot at $(4,\,2)$ , dipping to a lowest point of $y = -1$ in between.

Read the Sweep 🧮

A graph is a solid curve. Its leftmost point is a closed dot at $(-3,\,0)$ and its rightmost point is a closed dot at $(6,\,9)$ . Along the way it never goes below $y = 0$ .

1) Smallest $x$ -value in the domain $= \,?$ 2) Largest $x$ -value in the domain $= \,?$ 3) Smallest $y$ -value in the range $= \,?$

Functions with none of these — plain polynomials like $y = 3x^2 - x + 4$ — have domain $(-\infty,\, \infty)$ . Nothing can break them.

💡 Quick test: No fraction with $x$ on the bottom and no square root of $x$ ? Then the domain is all real numbers.

Concept Check 🎯

Fractions: Exclude Where the Bottom Is Zero

Example: $f(x) = \dfrac{1}{x - 3}$

Set the denominator equal to zero and solve:

$x - 3 = 0 \;\Rightarrow\; x = 3$

So $x = 3$ is forbidden; everything else is fine.

Inequality: $x \ne 3$
Interval: $(-\infty,\, 3) \cup (3,\, \infty)$ ← the symbol means "union" (glue the two pieces together).

Example: $g(x) = \dfrac{x+1}{x^2 - 4}$

$x^2 - 4 = 0 \;\Rightarrow\; x^2 = 4 \;\Rightarrow\; x = \pm 2$

Exclude both: domain is $x \ne 2$ and $x \ne -2$ .

⚠️ You only solve the denominator $= 0$ . The numerator never restricts the domain — it's perfectly fine for the top to be zero.

Find the Forbidden Inputs 🧮

For each function, enter the $x$ -value(s) that must be excluded from the domain.

1) $f(x) = \dfrac{4}{x - 8}$ . Excluded value $= \,?$ 2) $g(x) = \dfrac{x}{x + 5}$ . Excluded value $= \,?$ 3) $h(x) = \dfrac{6}{2x - 10}$ . Excluded value $= \,?$

Square Roots: Keep the Inside $\ge 0$

For an even root, force the radicand to be zero or positive.

Example: $f(x) = \sqrt{x - 2}$

$x - 2 \ge 0 \;\Rightarrow\; x \ge 2$

Domain: $[2,\, \infty)$ . (The endpoint $2$ is allowed, because $\sqrt{0} = 0$ is perfectly legal — so a square bracket.)

Example: $g(x) = \sqrt{2x + 6}$

$2x + 6 \ge 0 \;\Rightarrow\; 2x \ge -6 \;\Rightarrow\; x \ge -3$

Domain: $[-3,\, \infty)$ .

💡 Why $\ge$ and not $>$ ? Zero under the root is fine — $\sqrt{0}=0$ . Only insides are illegal, so the boundary point is always .

Solve the Radicand 🔽

Find the domain of each square-root function.

💡 For a parabola in vertex form, the range starts (or ends) at the vertex's $y$ -value $k$ . Up → range goes up from $k$ . Down → range goes down from $k$ .

Concept Check 🎯

Real-World Domains: Let Context Decide

In an application, a value that is mathematically legal can still be physically impossible. Ask: can this quantity really be negative? a fraction? huge?

Example: filling a pool

A pool fills at $5$ gallons per minute, holding at most $300$ gallons. The volume is $V(t) = 5t$ .

Math alone would allow any $t$ , even negative.
Context: time can't be negative, and once the pool is full at $t = 60$ minutes ( $5 \cdot 60 = 300$ ), filling stops.
Realistic domain: $0 \le t \le 60$ , i.e. $[0,\, 60]$ .
Realistic range: $0 \le V \le 300$ , i.e. $[0,\, 300]$ .

⚠️ Counting quantities (people, cars, tickets) force a domain of whole numbers — that's a discrete domain, not a smooth interval.

Restrict for Reality 🔽

A taxi charges a $3 flat fee plus $2 per mile: $C(m) = 3 + 2m$ , where $m$ is miles driven (up to a $50$ -mile limit).

Apply the Context 🧮

Use the pool from above: $V(t) = 5t$ , full at $300$ gallons, $0 \le t \le 60$ .

1) How many gallons after $t = 10$ minutes? $\,?$ 2) At what time $t$ is the pool full? $\,?$ 3) The largest value in the range of $V$ (in gallons) $= \,?$

⚠️ Bracket reminder: square $[\;]$ = included (matches $\le, \ge$ , closed dot), round $(\;)$ = excluded (matches $<, >$ , open dot, and always $\pm\infty$ ).

Mixed Practice 🎯

One More Set 🧮

1) Relation $\{(2,\,9),\;(5,\,9),\;(8,\,1)\}$ : how many distinct values are in the range? $\,?$ 2) $f(x) = \dfrac{1}{x + 6}$ : which $x$ -value is excluded from the domain? $\,?$ 3) $g(x) = \sqrt{x - 7}$ : what is the smallest $x$ in the domain? $\,?$

Exit Quiz ✅

Answer all three to finish the lesson.

$-2$	$2(-2)+1$	$-3$
$0$	$2(0)+1$	$1$
$3$	$2(3)+1$	$7$

Domain and Range

Domain and Range - Complete Interactive Lesson

Part 1: What Domain and Range Mean

🎯 Domain and Range

Topics in This Part

Inputs, Outputs, and Functions

Domain & Range of a Set of Points

Example

Three Ways to Describe a Set

Part 2: Reading Domain & Range from Graphs

🎯 Domain and Range

Project Onto the Axes

Part 3: Domain from an Equation

🎯 Domain and Range

The Two Forbidden Moves

Part 4: Range & Real-World Restrictions

🎯 Domain and Range

Range of Common Functions

Part 5: Mixed Practice & Mastery Check

🎯 Domain and Range

Quick Reference

Example: a parabola opening up

Open vs. Closed Endpoints

Example

Fractions: Exclude Where the Bottom Is Zero

Example: f(x)=1x−3f(x) = \dfrac{1}{x - 3}f(x)=x−31​

Example: g(x)=x+1x2−4g(x) = \dfrac{x+1}{x^2 - 4}g(x)=x2−4x+1​

Square Roots: Keep the Inside ≥0\ge 0≥0

Example: f(x)=x−2f(x) = \sqrt{x - 2}f(x)=x−2​

Example: g(x)=2x+6g(x) = \sqrt{2x + 6}g(x)=2x+6​

Real-World Domains: Let Context Decide

Example: filling a pool

Example: $f(x) = \dfrac{1}{x - 3}$

Example: $g(x) = \dfrac{x+1}{x^2 - 4}$

Square Roots: Keep the Inside $\ge 0$

Example: $f(x) = \sqrt{x - 2}$

Example: $g(x) = \sqrt{2x + 6}$