Domain and Range

🎯 Domain and Range

Part 1 of 5 — What Domain and Range Mean

Topics in This Part

🔑 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:

• 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.

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

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

• 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 $4$.

Read the Coordinates 🧮

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

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? $\,?$

Three Ways to Describe a Set

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

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.

Translate the Notation 🔽

Match each inequality to its correct interval.

🎯 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.

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 $[0,\, \infty)$.

💡 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) → $[1,\, 6)$.

⚠️ 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 $= \,?$

🎯 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

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 $\cup$ 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 negative insides are illegal, so the boundary point is always included.

Solve the Radicand 🔽

Find the domain of each square-root function.

🎯 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

💡 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) $= \,?$

🎯 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

⚠️ 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.