3.13Unit 3 Review: Functions & Linear Relationships
Bring together everything from Unit 3 — functions, function notation, domain and range, linear functions, and rate of change — in one comprehensive review and unit test.
Why This Matters
Functions and linear relationships are the most frequently tested topics on the SAT and ACT. A solid review here prepares you for systems of equations, quadratics, and every graphing topic in Algebra 2 and Precalculus.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Unit 3 Big Idea
A function is a rule that assigns exactly one output to each input. Linear functions have a constant rate of change (slope) and graph as straight lines. Slope, domain, range, and function notation are the tools that let you describe, evaluate, and compare linear relationships.
Chapter-by-Chapter Summary
Introduction to Functions
A function pairs each input with exactly one output. Use the vertical line test on graphs; check for repeated x-values in tables and sets.
Function Notation
f(x) is read "f of x" and means the output of function f at input x. Substitute the input value wherever you see x.
Domain and Range
Domain = all valid inputs (x-values). Range = all resulting outputs (y-values). State as a set, inequality, or interval notation.
Linear Functions
Linear functions have the form y = mx + b. Slope m gives the rate of change; b is the y-intercept. Graph by plotting the y-intercept and using slope to find more points.
Rate of Change & Slope
Slope = rise / run = (y₂ − y₁) / (x₂ − x₁). Find it from a graph, a table, or two points. Positive rises, negative falls, zero is horizontal, undefined is vertical.
Key Formulas & Rules — Unit 3 Quick Reference
Slope formula
m = (y₂ − y₁) / (x₂ − x₁)
Same subtraction order in numerator and denominator
Slope-intercept form
y = mx + b
m = slope, b = y-intercept
Function notation
f(x) = expression
f(a) means substitute x = a
Vertical line test
One intersection = function
Two or more = not a function
Domain (from graph)
All x-values the graph covers
Left to right; watch for endpoints
Range (from graph)
All y-values the graph covers
Bottom to top; watch for endpoints
Y-intercept
Set x = 0, solve for y
Point (0, b) on the graph
X-intercept
Set y = 0, solve for x
Point (a, 0) on the graph
Common Mistakes — Unit 3
- Confusing f(x) with multiplication: f(x) is not "f times x" — it means "the output of f at input x."
- Swapping domain and range: Domain = inputs (x); Range = outputs (y). They are not interchangeable.
- Mixing subtraction order in slope: Always subtract in the same order: (y₂ − y₁) over (x₂ − x₁). Swapping one flips the sign.
- Zero vs. undefined slope: Horizontal line → slope = 0. Vertical line → slope undefined. They are opposite errors.
- Forgetting the y-intercept when graphing: Always start at (0, b), then apply rise/run — don't start at the origin unless b = 0.
- Assuming every relation is a function: Always check — one x to two y-values fails the function definition.
- Treating slope as a point: m = 3 means "up 3, right 1" — it is a ratio describing direction, not a location.
Common Mistakes
Confusing domain (x-values) and range (y-values) when reading a graph or table.
Domain = all x-values (horizontal axis). Range = all y-values (vertical axis).
Reading f(x) as 'f times x' and trying to multiply.
f(x) means 'the output of f when the input is x.' It's function notation, not multiplication.
Plotting slope as run/rise instead of rise/run.
Slope = rise/run. Rise is vertical change; run is horizontal change.
Assuming every graph or table represents a function without checking the vertical line test or repeated x-values.
A relation is a function only if each x-value maps to exactly one y-value. Always verify.
Guided Practice
Guided Practice Video: Unit 3 Review
Watch the guided practice walkthrough for the Unit 3 review, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Answers are in the Answer Key section.
Determine whether each relation is a function: Set 1: {(1, 3), (2, 5), (3, 3), (4, 7)} — Set 2: {(1, 2), (1, 4), (2, 6)}
Hint: Check the x-values. In Set 1 every x-value (1, 2, 3, 4) is different — the repeated y-value 3 is fine, so it IS a function. In Set 2 the input x = 1 maps to both 2 and 4 — one input with two outputs is NOT a function.
Given f(x) = 3x − 5, find f(4) and f(−2).
Hint: Substitute each input for x. f(4) = 3(4) − 5 = 12 − 5 = 7. Use parentheses for the negative input: f(−2) = 3(−2) − 5 = −6 − 5 = −11.
State the domain and range of {(−2, 4), (0, 1), (3, 7), (5, −2)} in set notation.
Hint: Domain = all first coordinates: {−2, 0, 3, 5}. Range = all second coordinates, listed in ascending order: {−2, 1, 4, 7}.
A table shows: x = 0, 1, 2, 3 and y = −1, 2, 5, 8. Write the equation of the linear function.
Hint: y changes by +3 at every step, so m = 3. The x = 0 row gives the y-intercept: b = −1. Equation: y = 3x − 1.
Find the slope of the line through (2, 5) and (6, 13). Classify the slope.
Hint: m = (13 − 5) / (6 − 2) = 8 / 4 = 2. The slope is positive — the line rises from left to right.
Mixed Review — All Chapters
Ch 01 — Functions
Determine whether each relation is a function: {(1,3),(2,5),(3,3),(4,7)} and {(1,2),(1,4),(2,6)}.
A mapping diagram shows: 1→4, 2→7, 3→4, 4→9. Is this a function? Explain.
Use the vertical line test to determine whether a circle centered at the origin is a function.
Give an example of a real-world situation that is a function and one that is not.
Ch 02 — Function Notation
Given f(x) = 3x − 5, find: f(0), f(4), f(−2), and f(a + 1).
Given g(x) = x² − 2x + 1, find g(3) and g(−1).
Given h(x) = −2x + 7, solve h(x) = 1 and h(x) = −3.
A function is defined by the table: x = 0,1,2,3; f(x) = 5,8,11,14. Find f(2) and solve f(x) = 14.
Ch 03 — Domain and Range
State the domain and range of the relation {(−2,4),(0,1),(3,7),(5,−2)} in set notation.
A linear function is graphed from (−3, −1) to (4, 6) with closed endpoints. State domain and range in interval notation.
State the domain of f(x) = 1/(x − 3). Explain why x = 3 is excluded.
A function has domain [−2, 5] and range [0, 8]. Sketch a possible graph.
Ch 04 — Linear Functions
Graph y = 2x − 3. Identify the slope and y-intercept. Find the x-intercept.
Graph y = −(1/2)x + 4. Identify the slope and y-intercept. Find the x-intercept.
Write the equation of a line with slope 3 and y-intercept −2.
A table shows: x = 0,1,2,3; y = −1,2,5,8. Write the equation of the linear function.
A phone plan charges $30 per month plus $0.10 per text. Write a linear function for monthly cost C after t texts. What is the slope and what does it represent?
Ch 05 — Rate of Change & Slope
Find the slope through (2, 5) and (6, 13). Classify the slope.
Find the slope through (−3, 4) and (5, −4). Classify the slope.
Find the slope from the table: x = 0,2,4,6; y = 3,9,15,21.
Compare: Line A through (0,0) and (4,12); Line B through (0,0) and (4,8). Which has a greater rate of change?
A car travels 250 miles in 5 hours. What is the rate of change? Write a function and interpret the slope.
Cumulative Challenge Problems
A function f is defined by f(x) = 2x + 1. (a) Find f(3) and f(−2). (b) Find the slope of the graph of f. (c) State the domain and range. (d) Graph f.
A table shows: x = −2,0,2,4; y = 7,3,−1,−5. (a) Verify this is a linear function. (b) Find the slope. (c) Write the equation. (d) Find f(10).
Line A has equation y = 3x − 1. Line B passes through (0, 4) and (2, 0). (a) Find the slope of each line. (b) Which line is steeper? (c) Do the lines intersect? If so, where?
A water tank is being filled. After 2 minutes it holds 40 gallons; after 5 minutes it holds 100 gallons. (a) Find the rate of change. (b) Write a linear function for gallons G after t minutes. (c) How long until the tank holds 250 gallons?
★ A function f has slope 4 and passes through (1, 6). A function g has slope −2 and passes through (1, 6). (a) Write both equations. (b) Find the x-intercept of each. (c) At what x-value is f(x) = g(x)?