Piecewise, domain, range - ©R _ 2 F 0 V 2 w 2 g uK`uKteaB WSEo ... - Free Printable

Let's solve each problem step by step, analyzing the graphs to determine the domain and range in both set-builder notation and interval notation.

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🔹 Problem 1

#### Graph Description:
- A piecewise graph starting at $ x = -2 $ with a solid dot, going up to $ y = 3 $ at $ x = 0 $, then flat until $ x = 2 $, then decreasing linearly.
- The graph continues downward beyond $ x = 2 $, extending to infinity (no endpoint shown), but it goes through $ (6, -2) $, and appears to go further down.
- The leftmost point is at $ x = -2 $ (inclusive).
- The right side extends infinitely to the right (arrow indicates continuation).

So:
- Domain: All $ x $ from $ -2 $ to $ \infty $
- Range: From the lowest $ y $ value to highest. Highest is $ y = 3 $, and it decreases without bound → so $ (-\infty, 3] $

But wait — does it go below $ y = -\infty $? Yes, since it keeps going down.

Let’s examine:
- Starts at $ (-2, 1) $, goes up to $ (0,3) $, stays at $ y=3 $ to $ (2,3) $, then decreases.
- At $ x = 6 $, $ y = -2 $, and it continues decreasing.

So range is all $ y \leq 3 $, i.e., $ (-\infty, 3] $

✔ Domain:
- Starts at $ x = -2 $ (closed circle), goes to $ \infty $
- So: $ [-2, \infty) $

✔ Range:
- Maximum $ y = 3 $, and goes down to $ -\infty $
- So: $ (-\infty, 3] $

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#### ✔ Answers for Problem 1:

Setbuilder Notation:
- Domain: $ \{x \mid x \geq -2\} $
- Range: $ \{y \mid y \leq 3\} $

Interval Notation:
- Domain: $ [-2, \infty) $
- Range: $ (-\infty, 3] $

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🔹 Problem 2

#### Graph Description:
- A parabola-like shape, but not symmetric — starts at $ x = -2 $ (closed dot), goes down to $ y = -4 $ at $ x = 1 $, then up to $ y = 5 $ at $ x = 4 $, then down to $ x = 6 $ (open circle).
- The left end is at $ x = -2 $, closed.
- The right end is at $ x = 6 $, open (not included).
- So domain: $ [-2, 6) $
- Minimum $ y = -4 $, maximum $ y = 5 $
- So range: $ [-4, 5] $

Wait — is $ y = 5 $ achieved? Yes, at $ x = 4 $. Is $ y = -4 $ achieved? Yes, at $ x = 1 $. And all values in between are covered.

So:

✔ Domain: $ [-2, 6) $
✔ Range: $ [-4, 5] $

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#### ✔ Answers for Problem 2:

Setbuilder Notation:
- Domain: $ \{x \mid -2 \leq x < 6\} $
- Range: $ \{y \mid -4 \leq y \leq 5\} $

Interval Notation:
- Domain: $ [-2, 6) $
- Range: $ [-4, 5] $

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🔹 Problem 3

#### Graph Description:
- Two parts:
1. A single point at $ (-3, 4) $ — solid dot.
2. A line segment from $ (-3, -1) $ to $ (0, 3) $, but open at $ x = 0 $ (so not including $ (0,3) $).
- So includes $ x = -3 $ on the segment? Wait — at $ x = -3 $, there’s already a point at $ (-3,4) $, and the line segment starts at $ (-3, -1) $, which is different.

Wait — this is two separate pieces:
- One isolated point: $ (-3, 4) $
- A line segment from $ (-3, -1) $ to $ (0, 3) $, but open at $ x = 0 $, so $ x \in [-3, 0) $

But at $ x = -3 $, we have two points? No — check:
- The point $ (-3, 4) $ is above the line segment.
- The line segment starts at $ (-3, -1) $, which is a different point at same $ x $, lower $ y $.

So:
- At $ x = -3 $, two points: $ (-3, 4) $ and $ (-3, -1) $
- But is that allowed? Yes, because they’re different $ y $-values — function can have multiple outputs only if not single-valued, but here it’s a piecewise graph, so possibly not a function?

Wait — is this a function? Let's check vertical line test.

At $ x = -3 $, two $ y $-values: $ y = 4 $ and $ y = -1 $ → fails vertical line test → not a function?

But the worksheet says “function given by each graph” — so perhaps this is not a function, or maybe I misread.

Wait — look again.

The line segment starts at $ (-3, -1) $, but is it connected to the point $ (-3,4) $? No — they are separate.

But same $ x $, two $ y $'s → violates definition of function.

But the graph shows:
- A solid dot at $ (-3, 4) $
- A solid dot at $ (-3, -1) $
- Then a line segment from $ (-3, -1) $ to $ (0, 3) $, open at $ (0,3) $

So yes — two points at $ x = -3 $: $ y = 4 $ and $ y = -1 $ → not a function

But the worksheet says "function" — contradiction?

Possibly a typo or mistake? Or perhaps the point at $ (-3,4) $ is not part of the function? But it's drawn.

Wait — maybe the point at $ (-3,4) $ is separate, and the line segment starts at $ (-3, -1) $, but does not include $ (-3,4) $.

But still — two outputs for $ x = -3 $ → not a function.

Alternatively, maybe the point at $ (-3,4) $ is not connected, but it's part of the graph.

But unless specified otherwise, if a graph has more than one output for a single input, it's not a function.

But let's assume it's intended to be a function — maybe the point at $ (-3,4) $ is not meant to be at $ x = -3 $? But it clearly is.

Wait — looking closely:
- The point at $ (-3,4) $ is above the line segment.
- The line segment starts at $ (-3,-1) $, so at $ x = -3 $, two points.

Unless the point at $ (-3,4) $ is a single point, and the segment starts at $ (-3,-1) $, so both are defined at $ x = -3 $.

So unless it's a multivalued relation, this is not a function.

But since the worksheet says “function”, perhaps it's acceptable as a relation, but we'll proceed assuming it's a graph of a relation, and answer accordingly.

But the instruction says “function”, so likely an error.

Wait — perhaps the point at $ (-3,4) $ is not at $ x = -3 $? Let's check coordinates.

From grid:
- $ x = -3 $, $ y = 4 $: yes, solid dot
- Line segment: from $ (-3, -1) $ to $ (0, 3) $, open at $ (0,3) $

So both at $ x = -3 $: $ y = 4 $ and $ y = -1 $

→ Not a function

But let's assume it's a valid graph (maybe a typo), and compute domain and range anyway.

So:
- $ x $-values: $ x = -3 $ (from both points), and $ x \in [-3, 0) $ from the segment
- So total $ x $: $ [-3, 0) $ — includes $ x = -3 $, and goes to but not including $ x = 0 $

Wait — but the point at $ (-3,4) $ is already included in $ x = -3 $, and the segment also includes $ x = -3 $

So domain: $ [-3, 0) $

Now, range:
- $ y = 4 $ (from point)
- $ y $ from $ -1 $ to $ 3 $, not including $ 3 $
- So $ y \in [-1, 3) $, plus $ y = 4 $

So range: $ [-1, 3) \cup \{4\} $

Yes.

So:

✔ Domain: $ [-3, 0) $
✔ Range: $ [-1, 3) \cup \{4\} $

But again, this is not a function due to two outputs at $ x = -3 $

But since the problem says “function”, perhaps the point at $ (-3,4) $ is not part of the function? Or maybe it's a typo.

Alternatively, maybe the line segment starts at $ x = -3 $, but the point at $ (-3,4) $ is at a different location — but no, it's at $ x = -3 $

Wait — could the point at $ (-3,4) $ be not connected? But it's plotted.

I think we must accept it as is.

So even though it's not a function, we'll proceed.

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#### ✔ Answers for Problem 3:

Setbuilder Notation:
- Domain: $ \{x \mid -3 \leq x < 0\} $
- Range: $ \{y \mid -1 \leq y < 3\} \cup \{4\} $

Interval Notation:
- Domain: $ [-3, 0) $
- Range: $ [-1, 3) \cup \{4\} $

> Note: This is not a function due to two outputs at $ x = -3 $. Possibly a typo in the graph.

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🔹 Problem 4

#### Graph Description:
- A horizontal line at $ y = 3 $, extending infinitely left and right.
- Arrows on both ends → infinite in both directions.
- Solid line, no breaks.

So:
- Every real number $ x $ is included.
- $ y = 3 $ for all $ x $

So:
- Domain: All real numbers
- Range: Just $ y = 3 $

✔ Domain: $ (-\infty, \infty) $ or $ \mathbb{R} $
✔ Range: $ \{3\} $

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#### ✔ Answers for Problem 4:

Setbuilder Notation:
- Domain: $ \{x \mid x \in \mathbb{R}\} $ or $ \{x \mid -\infty < x < \infty\} $
- Range: $ \{y \mid y = 3\} $

Interval Notation:
- Domain: $ (-\infty, \infty) $
- Range: $ \{3\} $

> Note: In interval notation, a single value is written as $ \{3\} $, not an interval like $ [3,3] $, though sometimes people write $ [3,3] $, but standard is $ \{3\} $.

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## ✔ Final Answers

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1)

Setbuilder Notation:
- Domain: $ \{x \mid x \geq -2\} $
- Range: $ \{y \mid y \leq 3\} $

Interval Notation:
- Domain: $ [-2, \infty) $
- Range: $ (-\infty, 3] $

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2)

Setbuilder Notation:
- Domain: $ \{x \mid -2 \leq x < 6\} $
- Range: $ \{y \mid -4 \leq y \leq 5\} $

Interval Notation:
- Domain: $ [-2, 6) $
- Range: $ [-4, 5] $

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3)

Setbuilder Notation:
- Domain: $ \{x \mid -3 \leq x < 0\} $
- Range: $ \{y \mid -1 \leq y < 3\} \cup \{4\} $

Interval Notation:
- Domain: $ [-3, 0) $
- Range: $ [-1, 3) \cup \{4\} $

> ⚠️ Note: This graph is not a function because it has two outputs at $ x = -3 $. There may be an error in the graph.

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4)

Setbuilder Notation:
- Domain: $ \{x \mid x \in \mathbb{R}\} $
- Range: $ \{y \mid y = 3\} $

Interval Notation:
- Domain: $ (-\infty, \infty) $
- Range: $ \{3\} $

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