ActivitiesNew Math English Language Arts Science & Engineering Social Studies Global Languages Health & Physical Education Special Education Counseling & Life Skills Arts & Creativity ESL

Relations and Functions

A comprehensive Algebra II instructional unit on relations, function representations, domain and range, function notation, and operations and compositions.

Nntobin

Relations and Functions Practice Test

Algebra II Unit 1 Assessment Series

Relations & Functions Practice Test

Representations, Vertical Line Test, Domain & Range, Operations

Score

_____ / 60

Name:

Date:

Period:

Directions: Show all supporting work. Justify conclusions using algebraic definitions where required. Part 1: 16 pts

1

Function Identification & Relation Types

1. Relation $ R_1 $ (Mapping Diagram) [4 pts]

Determine whether the mapping represents a function. State if it is one-to-one.

Domain X -3 0 4 Range Y 2 5 9

Function? Yes No One-to-One? Yes No

Explanation:

2. Relation $ R_2 $ (Table Analysis) [4 pts]

Examine the table. Determine if it is a function. If not, state the value(s) of $ k $ that would make $ y $ a function of $ x $.

$ x $	-2	1	$ k $	5	7
$ y $	6	-3	4	6	0

If $ k = 1 $, is $ y $ a function of $ x $? Explain:

State all values $ k $ CANNOT be for $ R_2 $ to be a function:

$ k \neq $

3. Graph $ G_1 $ [4 pts]

Apply the Vertical Line Test. Classify and justify.

x y

Function? Yes No

Vertical Line Test Justification:

4. Graph $ G_2 $ [4 pts]

Apply the Vertical Line Test. Classify and justify.

x y

Function? Yes No

Vertical Line Test Justification:

Algebra II • Relations and Functions Practice Test Page 1 of 3

Algebra II Practice Test: Relations & Functions | Domain, Range & Function Notation

Name:

2

Domain and Range Analysis

Part 2: 12 pts

5. Continuous Function $ f(x) $ [6 pts]

x y -3 3 2 -2

State the domain and range using interval notation and inequality notation.

Domain (Interval):

D: [ )

Range (Interval):

R: [ )

Domain (Inequality):

Range (Inequality):

6. Algebraic Domain Restrictions [6 pts]

Find the domain of each function analytically. Express all final answers in interval notation.

a) $ g(x) = \frac{2x + 5}{x^2 - 4x - 12} $

Domain:

b) $ h(x) = \sqrt{3x - 15} $

Domain:

3

Function Notation & Evaluation

Part 3: 10 pts

Given the definitions: $ f(x) = -2x^2 + 3x - 1 $, $ p(x) = 4 - 3x $, and $ q(x) = |2x - 5| $

7. Evaluate $ f(-3) $ [3 pts]

$ f(-3) = $

8. Evaluate Expression [3 pts]

Find $ 2q(-1) - p(5) $

Answer:

9. Solve for $ x $ [4 pts]

Find $ x $ when $ p(x) = 19 $.

$ x = $

Algebra II • Relations and Functions Practice Test Page 2 of 3

Algebra II Practice Test: Relations & Functions | Operations & Modeling

Name:

4

Function Operations & Composition

Part 4: 12 pts

Given: $ f(x) = 2x - 3 $, $ g(x) = x^2 - 4x + 1 $, and $ h(x) = \frac{x-1}{x+2} $

10. Operations on Functions [3 pts]

Find and simplify: $ (f - g)(x) $

Answer:

11. Composite Expression [4 pts]

Find $ (g \circ f)(x) $ and expand standard form.

$ (g \circ f)(x) = $

12. Evaluate Composite [2 pts]

Find the value of $ (f \circ g)(-2) $.

$ (f \circ g)(-2) = $

13. Domain of Quotient [3 pts]

State all restrictions for $ \left(\frac{f}{h}\right)(x) $.

Restrictions: $ x \neq $

5

Real-World Application & Modeling

Part 5: 10 pts

14. Supply Chain Manufacturing Model [10 pts]

A robotics firm produces units per day based on operating hours $ t $ via $ N(t) = 45t $ ($ 0 \le t \le 12 $). Total cost (in dollars) for $ n $ units is modeled by $ C(n) = 120n + 3500 $.

a) Find a composite function $ C(N(t)) $ for total cost in terms of operating hours $ t $. [3 pts]

$ C(N(t)) = $

b) Evaluate & Interpret $ C(N(8)) $. [4 pts]

Meaning with units:

c) Practical Domain & Range [3 pts]

State in interval notation:

Domain:

Range:

Part 1: ___/16 Part 2: ___/12 Part 3: ___/10 Part 4: ___/12 Part 5: ___/10

Total Score: ______ / 60

Algebra II • Relations and Functions Practice Test Page 3 of 3 • End of Test

Relations and Functions Answer Key

Teacher Key Algebra II • Assessment Scoring Guide

Relations & Functions Practice Test

Full Step-by-Step Solutions, Point Distributions & Common Pitfalls

Total Points

60 Points

1

Part 1: Function Identification & Relation Types (16 Pts)

1. Relation $ R_1 $ (Mapping Diagram) 4 Pts

Function: ✓ YES (1 pt)

One-to-One: ✗ NO (1 pt)

Model Explanation:

It is a function because each input in Domain $ X $ maps to exactly one output in Range $ Y $. It is not one-to-one because two distinct inputs, $ 0 $ and $ 4 $, both map to the same output, $ 5 $ (many-to-one). (2 pts)

Pitfall: Students often mistake multiple inputs going to one output as a non-function. Remind them only repeating inputs with differing outputs violate functions.

2. Relation $ R_2 $ (Table Analysis) 4 Pts

If $ k = 1 $: NOT A FUNCTION

The input $ x = 1 $ would be assigned two different outputs: $ y = -3 $ and $ y = 4 $. A single domain element cannot yield multiple range values. (2 pts)

Disallowed values for $ k $:

$ k \neq -2, 1, 5, 7 $

$ k $ cannot equal any existing $ x $-value since its associated output is $ y = 4 $. (2 pts)

Scoring Note: Award 1 pt if student identifies at least 2 correct values, full 2 pts for all four.

3. Graph $ G_1 $ (Parabola) 4 Pts

Classification: ✓ IS A FUNCTION (1 pt)

Vertical Line Test Justification:

Any vertical line $ x = c $ drawn through the domain intersects the parabola at at most one point. Therefore, every domain input corresponds to a unique output $ y $. (3 pts)

Rubric: 1 pt for "at most one point of intersection"; 2 pts for connecting to single output per input.

4. Graph $ G_2 $ (Ellipse) 4 Pts

Classification: ✗ NOT A FUNCTION (1 pt)

Vertical Line Test Justification:

A vertical line (such as the y-axis, $ x = 0 $) intersects the ellipse at two distinct points (e.g. $ (0, 2) $ and $ (0, -2) $). This proves a single input yields more than one output. (3 pts)

Rubric: Full credit requires stating that vertical lines intersect more than once or providing a concrete counterexample.

Mastery Checklist for Part 1:

• Clear distinction between relation (any set of pairs) and function (unique output per input).

• Fluency with both algebraic and geometric criteria (Vertical Line Test).

Algebra II • Relations & Functions Answer Key Page 1 of 3

Algebra II Answer Key: Relations & Functions | Domain, Range & Function Notation

Part 2: 12 pts | Part 3: 10 pts

5. Continuous Function $ f(x) $ Domain & Range 6 Pts

Left endpoint: solid dot at $ (-4, -2) $. Right endpoint: open circle at $ (4, \approx 2) $. Global maximum: peak reaches $ y = 3 $.

Interval Notation (3 pts total)

Domain: [-4, 4) (1.5 pts: bracket at -4, paren at 4)

Range: [-2, 3] (1.5 pts: min -2, absolute max 3)

Inequality Notation (3 pts total)

Domain: -4 ≤ x < 4 (1.5 pts)

Range: -2 ≤ y ≤ 3 (1.5 pts)

6. Algebraic Domain Restrictions 6 Pts

a) Rational: $ g(x) = \frac{2x+5}{x^2 - 4x - 12} $ [3 pts]

1. Set denominator $ = 0 $: $ x^2 - 4x - 12 = 0 $

2. Factor: $ (x - 6)(x + 2) = 0 \implies x = 6, \; x = -2 $

3. Domain excludes zeros of denominator: $ x \neq -2, \; 6 $

Answer: $ (-\infty, -2) \cup (-2, 6) \cup (6, \infty) $

b) Radical: $ h(x) = \sqrt{3x - 15} $ [3 pts]

1. Set radicand $ \ge 0 $: $ 3x - 15 \ge 0 $

2. Add 15: $ 3x \ge 15 $

3. Divide by 3: $ x \ge 5 $

Answer: $ [5, \infty) $

3

Part 3: Function Notation & Evaluation (10 Pts)

7. Evaluate $ f(-3) $ 3 Pts

$ f(x) = -2x^2 + 3x - 1 $

f(-3) = -2(-3)² + 3(-3) - 1

= -2(9) - 9 - 1

= -18 - 9 - 1

= -28

$ f(-3) = $ -28

8. $ 2q(-1) - p(5) $ 3 Pts

$ p(x)=4-3x, \; q(x)=|2x-5| $

q(-1) = |2(-1)-5| = |-7| = 7

2q(-1) = 2(7) = 14

p(5) = 4 - 3(5) = -11

14 - (-11) = 14 + 11 = 25

Result = 25

9. Solve for $ x $ 4 Pts

$ p(x) = 19 \implies 4 - 3x = 19 $

4 - 3x = 19

-3x = 19 - 4

-3x = 15

x = 15 / -3 = -5

$ x = $ -5

Common Algebra II Evaluation Errors: In Problem 7, squaring negative numbers without parentheses ($-3^2 \neq 9$ vs $(-3)^2 = 9$). In Problem 8, dropping the double negative ($14 - (-11) = 25$, not $14 - 11 = 3$).

Algebra II • Relations & Functions Answer Key Page 2 of 3

Algebra II Answer Key: Relations & Functions | Operations, Compositions & Modeling

Part 4: 12 pts | Part 5: 10 pts

4

Part 4: Function Operations & Composition (12 Pts)

10. $ (f - g)(x) $ 3 Pts

(f - g)(x) = (2x - 3) - (x² - 4x + 1)

= 2x - 3 - x² + 4x - 1

= -x² + 6x - 4

Answer: -x² + 6x - 4

11. $ (g \circ f)(x) $ 4 Pts

g(f(x)) = (2x - 3)² - 4(2x - 3) + 1

= (4x² - 12x + 9) - 8x + 12 + 1

= 4x² - 20x + 22

Answer: 4x² - 20x + 22

12. $ (f \circ g)(-2) $ 2 Pts

1. g(-2) = (-2)² - 4(-2) + 1 = 4 + 8 + 1 = 13

2. f(13) = 2(13) - 3 = 26 - 3 = 23

Value: 23

13. Quotient Domain Restrictions 3 Pts

• $ h(x) $ denominator zero: $ x + 2 = 0 \implies x = -2 $

• $ h(x) = 0 $ (entire denominator): $ x - 1 = 0 \implies x = 1 $

Restrictions: $ x \neq -2, \; 1 $

5

Part 5: Real-World Modeling Solutions (10 Pts)

14. Supply Chain Manufacturing Model 10 Pts

a) Composite Cost Function $ C(N(t)) $ [3 pts]

Substitute $ N(t) = 45t $ into $ C(n) = 120n + 3500 $:

$ C(N(t)) = 120(45t) + 3500 = 5400t + 3500 $

b) Evaluate & Context [4 pts]

$ C(N(8)) = 5400(8) + 3500 = \$46,700 $ (2 pts)

Meaning with units: (2 pts)

"Operating the factory for 8 hours produces 360 units at a total production cost of $46,700."

c) Practical Domain & Range [3 pts]

Domain: [0, 12] (hours)

Range: [3500, 68300] ($)

$ C(0) = \$3,500 $; $ C(12) = 5400(12)+3500 = \$68,300 $

Suggested Grading Scale: A: 54-60 (90%+) • B: 48-53 (80-89%) • C: 42-47 (70-79%) • D: 36-41 (60-69%)

Pass Standard: 42+ / 60

Algebra II • Relations & Functions Answer Key Page 3 of 3 • End of Guide

You might also like

ILEARN Checkpoint Mastery

A multi-part Indiana Academic Standards Grade 3 Math ILEARN checkpoint preparation sequence featuring comprehensive 15-day daily spiral reviews and teacher answer keys for Checkpoints 2 and 3.

JQ

Janelle Quinn

4

Slope Sprint Launch

A focused algebra lesson on graphing linear equations of the form y = mx through the origin using slope as rise over run. Includes student guided practice and a complete teacher reference answer key.

CK

Catherine Kio

2

Slope Shifters Graphing Worksheet

A comprehensive three-page graphing worksheet focused on direct variation equations of the form y = mx. Features large, high-clarity coordinate planes with distinct 1-unit integer increments, structured tables of values, slope breakdown analysis, and a dedicated third page targeting the rise-over-run graphing method directly from the origin.

CK

Catherine Kio

Multiplication Strategy Lab

A comprehensive third-grade instructional sequence focusing on four foundational multiplication strategies: equal groups and arrays, repeated addition, skip counting patterns, and breaking apart factors. Each lesson pairs an actionable teacher guide with a targeted student practice worksheet.

JS

Jami Sharp

8

Functions and Graphs Review

A comprehensive high school algebra review covering function characteristics, linear models, inequalities, and absolute value graphs. Features targeted multi-part practice worksheets and an extensive teacher solutions guide.

N

ntobin

3

Cross Cross Answer Key

A two-page comprehensive teacher worked answer key for the Cross-Cross Warm Up, providing fully solved organizers, crossing movement diagrams, sign flip alerts, solution quantity checks, and completed number line graphs for both the equation and inequality.

H

Harlston

Pre-Algebra Concepts

A comprehensive pre-algebra and introductory geometry sequence covering linear equations, slope analysis, and coordinate plane transformations.

FM

Fernando Mancini

10

Checkpoint 3 Countdown

A 15-day Indiana Academic Standards Grade 3 Math Checkpoint 3 daily spiral review unit. Covers fractions, measurement, elapsed time, area, and data analysis through 15 student daily review pages and a complete teacher answer key.

JQ

Janelle Quinn

2

October 27th Warm Up: Cross Cross Inequalities Student Handout

A two-page comparative handout featuring the Boxes and Circles cross-cross organizer, with separated rows for Number of Solutions and Number Line Plot in the final solution cards across both pages.

H

Harlston

Ten Leap Intervention

A targeted two-week Tier 2 mathematics intervention focusing on standard 1.NBT.C.5. Students develop conceptual understanding and mental fluency finding ten more and ten less than any two-digit number.

K

kkipperman

7

Checkpoint Countdown

A comprehensive 15-day Indiana Academic Standards Grade 3 Math Checkpoint 2 ILEARN daily spiral review unit. Includes 15 student review pages with 6 daily problems and a complete educator answer key with standards alignment and grading guidance.

JQ

Janelle Quinn

2

Slope Sprint Answer Key

A complete 2-page teacher answer key and scoring reference for the Slope Sprint Worksheet. Features rendered solution lines on every coordinate grid, filled-in slope breakdown values, complete model responses for conceptual questions, and common misconception notes.

CK