Unit 1 · 11th Grade Mathematics
Linear Functions
& Applications.
Study guide covering representations, inverse functions, systems, and piecewise functions.
Topic A: Features of Linear Functions
Lessons 1–4 · Representations, inverse functions, and domain constraints.
01 — Lesson 1
Multiple Representations of Linear Functions
A linear function can be expressed as a graph, table, equation, or contextual description — each highlights different features.
- Graph: shows slope and intercepts visually.
- Table: constant rate of change between each row.
- Equation (y = mx + b): slope m, y-intercept b.
- Context: identifies meaning of variables and rate.
Example: f(x) = 3x + 5 → slope 3 (rises 3 per unit), y-intercept 5 (starts at 5).
Practice — Lesson 1
Q1 · Multiple Choice
Which representation of a linear function most clearly shows the rate of change?
Q2 · Short Answer
For f(x) = −2x + 7, what is the slope?
02 — Lesson 2
Inverse of a Linear Function
The inverse swaps input and output. If f(a) = b, then f⁻¹(b) = a.
- To find the inverse: swap x and y, then solve for y.
- Notation: f⁻¹(x) — read "f inverse of x."
- Graph: the inverse is a reflection over y = x.
- Context: if f converts miles to km, f⁻¹ converts km back to miles.
f(x) = 2x + 4 → swap: x = 2y + 4 → solve: y = (x−4)/2 → f⁻¹(x) = (x−4)/2
Practice — Lesson 2
Q3 · Short Answer
Find the inverse of f(x) = 3x − 6. Write your answer as f⁻¹(x) = ...
Q4 · Multiple Choice
If f(5) = 12, what is f⁻¹(12)?
03 — Lesson 3
Domain Constraints in Context
Real-world situations limit the valid inputs of a function — these are domain constraints.
- Constraint: a restriction on domain or range based on context.
- You can't have negative hours, negative prices, or more than 100%.
- Write constraints as inequalities: 0 ≤ x ≤ 40.
- Extraneous solutions: algebraic answers that violate constraints.
A worker earns $15/hr: f(h) = 15h. Constraint: 0 ≤ h ≤ 40 (can't work negative or infinite hours).
Practice — Lesson 3
Q5 · Multiple Choice
A parking garage charges $4/hour, open 7am–10pm. Which constraint is correct for hours parked (h)?
Q6 · Multiple Choice
An algebraic solution gives x = −3 hours. In context, this solution is:
04 — Lesson 4
Composition of Functions
Composing functions means using the output of one function as the input of another.
- Notation: (f ∘ g)(x) = f(g(x)) — apply g first, then f.
- Order matters: f(g(x)) ≠ g(f(x)) in general.
- Context: a tip calculated on a discounted price — discount first, then tip.
f(x) = 2x, g(x) = x + 3
(f ∘ g)(x) = f(x+3) = 2(x+3) = 2x + 6
Practice — Lesson 4
Q7 · Short Answer
If f(x) = x + 5 and g(x) = 3x, find (g ∘ f)(2). (Apply f first, then g.)
Q8 · Multiple Choice
f(x) = x − 2, g(x) = 4x. What is (f ∘ g)(x)?
Topic B: Systems of Functions & Constraints
Lessons 5–9 · Linear systems, three-variable systems, absolute value.
05 — Lesson 5
Systems of Two Linear Equations
A system is two or more equations solved simultaneously. The solution is where they intersect.
- Substitution: isolate one variable, plug into the other equation.
- Elimination: add or subtract equations to cancel a variable.
- Graphically: the intersection point is the solution.
- No solution: parallel lines. Infinite solutions: same line.
2x + y = 8 and x − y = 1 → Add: 3x = 9 → x = 3 → y = 2. Solution: (3, 2)
Practice — Lesson 5
Q9 · Multiple Choice
Two linear equations have no solution. What does this mean graphically?
Q10 · Short Answer
Solve the system: y = 2x − 1 and y = x + 3. What is x?
06 — Lesson 6
Systems with Three Variables
Extend systems to three equations and three unknowns. Solve by eliminating one variable at a time.
- Step 1: Use two pairs of equations to eliminate the same variable.
- Step 2: Solve the resulting 2-variable system.
- Step 3: Back-substitute to find the third variable.
- Solution is an ordered triple: (x, y, z).
The solution to a 3-variable system represents a point where three planes intersect in 3D space.
Practice — Lesson 6
Q11 · Multiple Choice
What is the form of a solution to a 3-variable linear system?
Q12 · Multiple Choice
When solving a 3-variable system, what is the first goal?
07–09 — Lessons 7–9
Absolute Value Functions & Systems
Absolute value creates a V-shaped graph. It can appear in functions and in systems.
- |x|: returns the non-negative distance from zero. |−3| = 3.
- f(x) = |x − h| + k: vertex at (h, k), opens upward.
- Solving |x| = a: gives x = a or x = −a (two cases).
- System with abs. value: find where |expression| intersects another function.
|x − 2| = 5 → x − 2 = 5 or x − 2 = −5 → x = 7 or −3
Practice — Lessons 7–9
Q13 · Short Answer
Solve: |2x − 4| = 10. Give the larger solution.
Q14 · Multiple Choice
The graph of f(x) = |x| has what shape?
Topic C: Piecewise Functions
Lessons 10–13 · Defining, analyzing, and modeling with piecewise functions.
10 — Lesson 10
Defining Piecewise Functions
A piecewise function uses different rules for different intervals of the domain.
- Each "piece" is a separate function valid only on a specified interval.
- Notation: written with a brace { listing each rule and its domain interval.
- Evaluating: identify which interval x falls in, then use that rule.
- The pieces happen in sequence — not simultaneously like a system.
f(x) = { 2x if x < 3 ; x + 5 if x ≥ 3 }
f(1) = 2(1) = 2 · f(4) = 4 + 5 = 9
Practice — Lesson 10
Q15 · Short Answer
Using f(x) = { 3x if x ≤ 2 ; x + 8 if x > 2 }, what is f(5)?
Q16 · Multiple Choice
A piecewise function has two "pieces." When evaluating f(−1), how do you decide which piece to use?
11–13 — Lessons 11–13
Analyzing & Modeling with Piecewise Functions
Use piecewise functions to model real situations where the rule changes at a certain threshold.
- Tax brackets: different rates for different income ranges.
- Shipping costs: flat rate up to a weight, then per-pound beyond.
- Continuity: check that pieces "connect" — no gaps or jumps — at breakpoints.
- Graph: use open/closed circles to show whether endpoints are included.
A phone plan: $30/mo for 0–2 GB, $30 + $10(x−2) for x > 2 GB → piecewise model.
Practice — Lessons 11–13
Q17 · Multiple Choice
A piecewise function has an open circle at (3, 7) and a closed circle at (3, 5). What does this mean?
Q18 · Multiple Choice
Which real-world situation is best modeled by a piecewise function?