Domain and Range Introduction | Solved Exercises

Solution: Exercises 1 to 3

1 Domain and Range from Graphs

Exercise 1

From the graph of function f(x), identify the domain and range. The graph shows a parabola opening upward with vertex at (2, -3) and extending indefinitely in both horizontal directions.

Definition:

Domain: All possible x-values (horizontal extent) where the function is defined

Range: All possible y-values (vertical extent) that the function outputs

Method for finding domain and range from graphs:

Examine the horizontal extent of the graph (left to right) for domain
Examine the vertical extent of the graph (bottom to top) for range
Look for any breaks, holes, or asymptotes that limit the function
Express the result using interval notation

Graph Features

Parabola, vertex (2, -3)

Domain

All real numbers

Range

y ≥ -3

Step 1: Analyze the domain

Looking horizontally: The parabola extends infinitely to the left and right

No x-values are excluded

Domain: (-∞, ∞) or all real numbers

Step 2: Analyze the range

Looking vertically: The lowest point is the vertex at y = -3

The parabola opens upward, so y-values increase from -3

All y-values ≥ -3 are possible

Range: [-3, ∞)

Step 3: Express in interval notation

Domain: (-∞, ∞) - parentheses because infinity is not included

Range: [-3, ∞) - bracket at -3 because it's included, parenthesis at ∞

Step 4: Verify with function characteristics

For f(x) = (x - 2)² - 3 (based on vertex form):

Domain is all real numbers (no restrictions)

Range is y ≥ -3 (vertex is minimum point)

Domain: (-∞, ∞)
Range: [-3, ∞)

Final answer:

Domain: (-∞, ∞) or all real numbers

Range: [-3, ∞) or y ≥ -3

Applied rules:

• Graph analysis: Domain = horizontal extent, Range = vertical extent

• Interval notation: Parentheses exclude endpoints, brackets include endpoints

• Parabola properties: Opens upward → minimum value, opens downward → maximum value

2 Domain Restrictions

Exercise 2

Find the domain of f(x) = 1/(x - 4) and g(x) = √(x + 2). Explain why these restrictions exist.

Definition:

Domain restriction: Values of x that make the function undefined

Rational function: Function expressed as a fraction

Radical function: Function containing a square root

Function 1

f(x) = 1/(x - 4)

Function 2

g(x) = √(x + 2)

Domains

f: x ≠ 4, g: x ≥ -2

Step 1: Find domain of f(x) = 1/(x - 4)

Rational functions: denominator cannot equal zero

So: x - 4 ≠ 0

Therefore: x ≠ 4

Domain: (-∞, 4) ∪ (4, ∞)

Step 2: Find domain of g(x) = √(x + 2)

Radical functions: expression under root must be ≥ 0

So: x + 2 ≥ 0

Therefore: x ≥ -2

Domain: [-2, ∞)

Step 3: Explain the restrictions

For f(x): Division by zero is undefined, so x cannot equal 4

For g(x): Negative numbers under square roots are not real, so x + 2 ≥ 0

Step 4: Express in different notations

f(x) domain: x ∈ ℝ, x ≠ 4 or {x | x ≠ 4}

g(x) domain: x ≥ -2 or [-2, ∞)

Step 5: Verify with examples

f(5) = 1/(5-4) = 1/1 = 1 ✓

f(4) = 1/(4-4) = 1/0 → undefined ✗

g(-2) = √(-2+2) = √0 = 0 ✓

g(-3) = √(-3+2) = √(-1) → not real ✗

f(x) domain: (-∞, 4) ∪ (4, ∞)
g(x) domain: [-2, ∞)

Final answer:

f(x) = 1/(x - 4): Domain is all real numbers except x = 4

Notation: (-∞, 4) ∪ (4, ∞) or {x ∈ ℝ | x ≠ 4}

g(x) = √(x + 2): Domain is all real numbers x ≥ -2

Notation: [-2, ∞) or {x ∈ ℝ | x ≥ -2}

Applied rules:

• Rational functions: Denominator ≠ 0

• Radical functions: Expression under √ ≥ 0

• Domain restrictions: Always check for undefined expressions

3 Range Determination

Exercise 3

Find the range of f(x) = x² - 4x + 3. Identify the vertex and explain how it determines the range.

Definition:

Range: Set of all possible output values (y-values)

Vertex form: f(x) = a(x - h)² + k, where (h, k) is the vertex

Function

f(x) = x² - 4x + 3

Vertex form

f(x) = (x - 2)² - 1

Range

y ≥ -1

Step 1: Rewrite in vertex form by completing the square

f(x) = x² - 4x + 3

f(x) = x² - 4x + ? + 3 - ?

To complete the square: take coefficient of x (which is -4), divide by 2: -4/2 = -2, then square: (-2)² = 4

f(x) = x² - 4x + 4 + 3 - 4

f(x) = (x - 2)² - 1

Step 2: Identify the vertex

In vertex form f(x) = (x - h)² + k, the vertex is (h, k)

So for f(x) = (x - 2)² - 1, the vertex is (2, -1)

Step 3: Determine the range

Since the coefficient of (x - 2)² is positive (1 > 0), the parabola opens upward

The vertex (2, -1) is the minimum point

Therefore, the minimum y-value is -1

All y-values ≥ -1 are possible

Step 4: Express the range

Range: [-1, ∞) or y ≥ -1

Step 5: Verify with examples

At vertex x = 2: f(2) = (2-2)² - 1 = 0 - 1 = -1 (minimum value)

At x = 0: f(0) = (0-2)² - 1 = 4 - 1 = 3

At x = 4: f(4) = (4-2)² - 1 = 4 - 1 = 3

All calculated values are ≥ -1 ✓

Vertex: (2, -1)
Range: [-1, ∞)

Final answer:

Vertex: (2, -1)

Range: [-1, ∞) or y ≥ -1

Explanation: Since the parabola opens upward, the vertex represents the minimum value, and all y-values greater than or equal to -1 are possible outputs.

Applied rules:

• Completing the square: Method to convert standard to vertex form

• Parabola orientation: Positive coefficient → opens upward, negative → opens downward

• Vertex significance: Determines minimum (upward) or maximum (downward) value

Domain and Range: Complete Introduction Guide

f: X → Y

Function Mapping

Domain

Input values

Range

Output values

Interval

[a, b]

Bracket notation

Key definitions:

Domain: The set of all possible input values (x-values) for which a function is defined

Range: The set of all possible output values (y-values) that a function can produce

Function: A relation where each input has exactly one output

Interval notation: A way to represent sets of numbers using brackets and parentheses

Domain and range finding methodology:

From graphs: Examine horizontal (domain) and vertical (range) extents
From equations: Identify restrictions based on function type
For polynomials: Domain is typically all real numbers
For rationals: Exclude values that make denominator zero
For radicals: Ensure expression under root is non-negative
For range: Analyze function behavior and critical points

Tip 1: Domain = inputs (x-values), Range = outputs (y-values) - remember "DR" = "Domain, Range".

Tip 2: Always check for division by zero, negative square roots, and other undefined expressions.

Tip 3: For quadratic functions, find the vertex to determine the range.

Tip 4: Graph the function to visualize domain and range when in doubt.

Common errors: Forgetting domain restrictions, confusing domain and range, misinterpreting interval notation.

Key insights: Domain restrictions depend on function type, range depends on function behavior and domain.

Essential formulas and rules:

• Interval notation: [a, b] includes endpoints, (a, b) excludes endpoints

• Rational functions: Domain excludes zeros of denominator

• Radical functions: Domain requires expression under root ≥ 0

• Quadratic functions: Range determined by vertex and direction of opening

• Linear functions: Domain and range are typically all real numbers

Solution: Exercises 4 to 5

4 Domain and Range from Tables

Exercise 4

Given the function represented by the table below, identify the domain and range. Then determine if this could represent a continuous function.

Definition:

Discrete function: Function defined only at specific points

Continuous function: Function defined for all values in an interval

Table Data

x: {-2, 0, 1, 3, 5}, y: {4, 0, 1, 9, 25}

Domain

{-2, 0, 1, 3, 5}

Range

{0, 1, 4, 9, 25}

Step 1: Identify the domain from the table

Domain consists of all x-values in the table

Domain: {-2, 0, 1, 3, 5}

Step 2: Identify the range from the table

Range consists of all y-values in the table

Range: {0, 1, 4, 9, 25}

Step 3: Observe the pattern in the data

Notice: (-2)² = 4, (0)² = 0, (1)² = 1, (3)² = 9, (5)² = 25

This suggests f(x) = x² for these specific points

Step 4: Determine continuity possibility

Since the function is only defined at specific points, it's discrete

However, it could be part of a continuous function f(x) = x²

If extended to all real numbers, the continuous version would have:

Domain: (-∞, ∞) and Range: [0, ∞)

Step 5: Present the results

For the given table (discrete function):

Domain: {-2, 0, 1, 3, 5}

Range: {0, 1, 4, 9, 25}

Step 6: Compare with potential continuous function

If this table represents samples from f(x) = x²:

Continuous domain: (-∞, ∞)

Continuous range: [0, ∞)

Table domain: {-2, 0, 1, 3, 5}
Table range: {0, 1, 4, 9, 25}
Could represent continuous f(x) = x²

Final answer:

From the table:

Domain: {-2, 0, 1, 3, 5}

Range: {0, 1, 4, 9, 25}

Yes, this could represent a continuous function (specifically, samples from f(x) = x²).

Applied rules:

• Discrete vs continuous: Tables show discrete functions; equations can represent continuous functions

• Domain from table: Collect all x-values

• Range from table: Collect all y-values

5 Piecewise Function Domains and Ranges

Exercise 5

Find the domain and range of f(x) = { x + 1 if x < 0, x² if x ≥ 0 }. Sketch the graph and explain your reasoning.

Definition:

Piecewise function: A function defined by different expressions over different intervals

Domain union: The overall domain is the union of domains of all pieces

Piecewise Function

f(x) = {x + 1 if x < 0, x² if x ≥ 0}

Domain

All real numbers

Range

y ≥ 0

Step 1: Analyze the first piece: f(x) = x + 1 when x < 0

Domain for this piece: x < 0, or (-∞, 0)

As x approaches 0 from the left, f(x) approaches 1

As x approaches -∞, f(x) approaches -∞

Range for this piece: (-∞, 1)

Step 2: Analyze the second piece: f(x) = x² when x ≥ 0

Domain for this piece: x ≥ 0, or [0, ∞)

When x = 0, f(x) = 0

As x increases, f(x) increases without bound

Range for this piece: [0, ∞)

Step 3: Find the overall domain

The domain is the union of both pieces' domains

Overall domain: (-∞, 0) ∪ [0, ∞) = (-∞, ∞)

All real numbers are covered

Step 4: Find the overall range

Range is the union of both pieces' ranges

First piece range: (-∞, 1)

Second piece range: [0, ∞)

Combined range: (-∞, 1) ∪ [0, ∞) = (-∞, ∞)

Wait, let me reconsider...

Actually, the first piece gives values (-∞, 1) but only for x < 0

The second piece gives values [0, ∞) for x ≥ 0

So combined range is (-∞, 1) ∪ [0, ∞) = (-∞, ∞)

Step 5: Correct range analysis

Let me reconsider: For x < 0, f(x) = x + 1, so as x approaches 0⁻, f(x) approaches 1

For x ≥ 0, f(x) = x², so f(0) = 0

The function values go from -∞ up to (but not including) 1, then from 0 onwards

Since [0, ∞) includes 0 and overlaps with values approaching 1, the range is (-∞, 1) ∪ [0, ∞) = (-∞, ∞)

Step 6: Final verification

Actually, let me be more precise:

For x < 0: f(x) = x + 1, so f(x) < 1 (since x < 0, x + 1 < 1)

For x ≥ 0: f(x) = x², so f(x) ≥ 0

At the boundary x = 0: Using x ≥ 0 rule, f(0) = 0² = 0

The range is all values less than 1 combined with all values ≥ 0

This means range is (-∞, 1) ∪ [0, ∞) = (-∞, ∞)

Domain: (-∞, ∞)
Range: (-∞, ∞)

Final answer:

Domain: (-∞, ∞) or all real numbers

Range: (-∞, ∞) or all real numbers

Explanation: The first piece covers x < 0 with outputs approaching 1, and the second piece covers x ≥ 0 with outputs starting at 0 and going to infinity, together covering all real numbers.

Applied rules:

• Piecewise domain: Union of domains of all pieces

• Piecewise range: Union of ranges of all pieces

• Boundary considerations: Check which piece applies at boundary values

Advanced Domain and Range Concepts

\text{Domain} = \{x | f(x) \text{ is defined}\}

Domain Definition

Key definitions:

Function: A relation where each element in the domain maps to exactly one element in the range

Domain: The set of all possible input values (independent variable)

Range: The set of all possible output values (dependent variable)

Interval notation: A concise way to describe sets of real numbers using brackets and parentheses

Comprehensive domain and range analysis:

Identify function type: Polynomial, rational, radical, exponential, etc.
Look for restrictions: Division by zero, negative under radicals, etc.
Analyze behavior: End behavior, critical points, asymptotes
Consider domain: What x-values are allowed?
Determine range: What y-values can be achieved?
Express results: Use appropriate notation (set, interval, inequality)

Tip 1: Always check for domain restrictions first - these are the most common source of limitations.

Tip 2: For range, think about the minimum and maximum possible output values.

Tip 3: Graphing can provide visual confirmation of your analytical results.

Tip 4: Test boundary values and critical points to verify your findings.

Common misconceptions: Assuming all functions have unlimited domains, confusing domain and range, misapplying interval notation.

Memory aids: Domain = Inputs (x-values), Range = Outputs (y-values); "DR" = "Domain, Range".

Essential formulas and relationships:

• Interval notation: [a,b] closed interval, (a,b) open interval, [a,b) half-open

• Polynomial functions: Domain is all real numbers

• Rational functions: Exclude zeros of denominator from domain

• Radical functions: Expression under even root ≥ 0

• Quadratic functions: Range depends on vertex and direction of opening

Visualization: Domain and Range Examples

Exercise 6: Function Comparison

Compare the domains and ranges of:
f(x) = x²
g(x) = √x
h(x) = 1/x

Analysis: Different function types have different domain and range restrictions.

Quadratic: Domain all reals, Range y ≥ 0
Radical: Domain x ≥ 0, Range y ≥ 0
Rational: Domain x ≠ 0, Range y ≠ 0

Solved Exercises on Domain and Range Introduction

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