Comprehensive Study Notes: Functions, Domain/Range, Transformations, Inverses, and Linear Modeling

Functions and Transformations: Comprehensive Study Notes

These notes summarize key ideas, concepts, and typical problem types found across the transcript. Where items are garbled in the transcript, I provide the standard interpretation and clearly mark ambiguities.

Functions, Domain, and Range (Overview)

A relation assigns to each input a value (or values) of the output. A relation is a function if every input is associated with exactly one output (vertical line test).
Domain: set of all input values x for which the relation is defined.
Range: set of all output values y that actually occur from the domain inputs.
For a function given explicit domain, compute the range by applying the function to every domain value.
For a function given a range, determine the corresponding domain by solving for inputs that produce those outputs.
Special cases:
- Constant functions: domain is all allowed inputs (often all reals); range is a single value.
- Linear functions: typically domain and range are all real numbers unless restricted; slope determines whether function is increasing or decreasing.
- Absolute value, square root, and other non-linear forms: domains can be restricted by the square root (requiring nonnegative radicands), etc.
Notation reminders: use LaTeX for math expressions. Example: $y = x^2, ext{ domain } x \,\in\, {-4,-3,-2,-1,0,1,2,3,4}$ gives a range of the squared values.

Page 1: Domain and Range Practice

1) Function: $y = x^2,ext{domain } x \in\ { -4,-3,-2,-1,0,1,2,3,4 }$
- Range: ${0,1,4,9,16}$
2) Function: $m = 3\sqrt{n},\; n \ge 0$
- Domain: $n \ge 0$ (given) but typically domain is all nonnegative reals; Range: $m \ge 0$ ; in fact $m = 3\sqrt{n}$ maps to all nonnegative reals: $\text{Range } = [0, \infty).$
3) Function: $f(x) = x^4,\; x \in \mathbb{R}$
- Range: $f(x) \ge 0\Rightarrow [0, \infty).$
4) Function: $s = -|t-3|,\quad -10 \le t \le 10$
- Range: maximum at t = 3 gives 0; minimum at endpoints t = -10 or 10 gives $-13$ ; so $s \in [-13,\,0].$
5) Function: $p = \sqrt{r+3},\quad p \ge 3$
- Relationship between domain and range: If $p = \sqrt{r+3}$ and the range is restricted to $p \ge 3$ , then solve $\sqrt{r+3} \ge 3 \Rightarrow r+3 \ge 9 \Rightarrow r \ge 6.$ Domain for r: $[6, \infty).$
6) Function: $z = -2w^2,\quad z \le 0$
- Because $w^2 \ge 0\Rightarrow -2w^2 \le 0$ for all real w, the domain is all real numbers.
- Range: as w grows, z goes to $-\infty$; the maximum is at w = 0 giving $z=0$ ; hence $\text{Range } = (-\infty,\,0].$
7) Function: $y = 3x\;$ (note: transcript shows ambiguous spacing; if intended as $y = 3x$ , then)
- Domain: $\mathbb{R}$ ; Range: $\mathbb{R}$ .
- If the transcript meant a different form (e.g., exponential or something else), interpret accordingly; keep the standard result for a simple linear function unless domain restrictions are stated.
8) Function: $u(t) = |2 - t|,\quad 0 \le u(t) \le 4$
- From the range restriction, solve $0 \le |2 - t| \le 4$ ⇒ $|2 - t| \le 4$ ⇒ $-4 \le 2 - t \le 4$ ⇒ $-2 \le t \le 6.$ Domain: $t \in [-2,\,6].$
9) Graphs (problem 9): domain and range from graphs
- 9a: Graph shows a discrete domain (gaps between domain values). Domain is a discrete set of input values (not an interval). Range is the corresponding discrete set.
- 9b: Graph shows a continuous domain (an interval). Domain is an interval; range is an interval as well.
10) Conceptual difference (discrete vs continuous domains):
- Discrete domain: input values are separate, distinct points (no continuum between them).
- Continuous domain: inputs cover an entire interval; between any two domain values there are infinitely many others.

Page 2–3: Function Investigation (Domain, Range; Is it a Function?)

Core idea: for each relation, determine the domain (inputs) and range (outputs). Many items involve common function types (linear, constant, quadratic, square root, absolute value, etc.).
Typical results you should know:
- Linear relation f(x) = ax + b: Domain = all real numbers; Range = all real numbers.
- Constant function f(x) = c: Domain = all real numbers; Range = {c}.
- Quadratic function y = ax^2 + bx + c with a ≠ 0: Domain = all real numbers; Range depends on a and vertex; often [ymin, ∞) if a > 0, or (−∞, ymax] if a < 0, subject to domain restrictions.
- Square root function y = √(g(x)): Domain restricted so g(x) ≥ 0; Range y ≥ 0.
- Absolute value y = |h(x)|: Range y ≥ 0; Domain depends on h(x) if presented with restrictions.
- For relations involving two variables representing physical quantities (e.g., bags of mandarins with a capacity constraint), interpret domain and range in context (e.g., domain could be number of bags, range could be total mandarins).
Ambiguities in the transcript (garbled notation) appear in problems 5–9 and some others. When practicing, rely on standard function types and the rule that a relation is a function if every input has a unique output.
Quick worked example (clear from transcript):
- If a relation is f(x) = 7x − 1, then Domain = all real numbers, Range = all real numbers.
- If a relation is a constant h = −10, Domain = all real numbers, Range = {−10}.
- If a relation is p = √(r+3) with p ≥ 3, Domain for r is [6, ∞) (as shown above).

Page 4–5: Increasing and Decreasing (Definitions and Examples)

Definitions:
- Increasing on [a,b] if for any x1 < x2 in [a,b], f(x1) < f(x2).
- Decreasing on [a,b] if for any x1 < x2 in [a,b], f(x1) > f(x2).
Examples from transcripts (interpretations):
- y = x^2 on [0, ∞): Increasing on [0, ∞) (strictly increasing for x > 0).
- y = x^2 on [-5, 5]: Not increasing on the entire interval; it decreases on [-5,0] and increases on [0,5].
- y = x^3 on [0,5]: Increasing (since derivative 3x^2 > 0 for x > 0).
- y = |x − 1|: Increasing on [1, ∞); Decreasing on (-∞, 1].
For graphs with multiple pieces (as in transcripts), identify where the function goes up or down over specified subintervals.
Table-based questions (Page 5):
- If z(x) increases as x increases, the table represents an increasing function on its domain.
- If t decreases as x increases (or vice versa), note the monotonic relationship accordingly.

Page 6–7: Linear Functions and Properties; Linear Modeling

Part 1 (Plotting/identifying linear functions):
- Given equations of the form y = mx + b, identify slope m and intercept b; plot using two clear points if needed.
- Slopes: positive slope means increasing; negative slope means decreasing.
Part 2 (Through two points):
- Slope m = (y2 − y1)/(x2 − x1).
- Use slope-intercept form y = mx + b; solve for b using a known point.
Part 3 (Intercepts):
- x-intercept: set y = 0 and solve for x.
- y-intercept: set x = 0 and solve for y.
Part 4–6 (Writing equations from conditions):
- E.g., a line with a given slope that passes through a point; or a line parallel to a given line through a point; or a line perpendicular to a given line through a point.
Part 7–9 (Monotonicity of linear functions):
- Increasing: slope m > 0.
- Decreasing: slope m < 0.
Part 10–12 (Linear equations from graph or given conditions):
- Translate from graph (slope and intercepts) to equation form.
Part 13–15 (Increasing/Decreasing outcomes for specific lines):
- Examples show how to determine monotonicity from the slope (e.g., a line with positive slope is increasing across its domain).
Part 16–18 (Find the linear equation from a graph):
- Use x-intercept and y-intercept if provided; otherwise use two points on the line to solve for m and b.
Practical tips from these problems:
- For a line through (x1,y1) and (x2,y2): compute $m = \frac{y2 - y1}{x2 - x1}$ and then $b = y1 - m x1.$
- Intercepts give quick checks: x-intercept where y=0; y-intercept where x=0.

Page 7: Linear Modeling and Applications

Concept: Build linear models from data to describe relationships between two quantities (e.g., gas mileage vs. gallons; speed vs. time).
Example 1 (Felicia’s car):
- Data: gallons vs. miles (table provided).
- Part (a): Treat miles as a function of gallons; fit a line: slope ~ miles per gallon (efficiency).
- Part (b): Treat gallons as a function of miles; slope ~ gallons per mile (fuel needed per mile).
- Part (c): Compare the two models; discuss when each is useful (e.g., MPG vs gallons per mile).
- Part (d): Reasons data may not fit perfectly: traffic, driving conditions, human factors, measurement error, etc.
Example 2 (Speed vs. time in a fall):
- Linear relation: speed V(t) = at + b; for free fall with constant gravity (ignoring air resistance), a is the acceleration (in many classroom setups, a = −g or a = −10 m/s^2 in a simplified unit system).
- Slope represents speed change per unit time; y-intercept represents initial speed at t = 0.
- Negative slope reflects downward velocity if downward is taken as negative direction.
General takeaway: Slopes have units that match the dependent variable per unit of the independent variable; intercepts provide initial conditions.

Page 8–9: Working with the Absolute Value Function

Solve equations of the form a|expression| + b = c by isolating the absolute value and solving the two cases: expression = ±(c−b)/a.
Example 1: Solve 6|x − 3| = 48
- |x − 3| = 8
- x − 3 = ±8 ⇒ x = 11 or x = −5.
Example 2–3: More complex absolute-value equations (transcript contains garbled forms; approach remains case-splitting).
Graphing problems (4–6):
- Graph h(x) = 2|x − 1| + 1 and similar; intercepts found by solving h(x) = 0 or y = 0, etc.
Intercepts (problem 7): For a function like H(x) = |2x − 1| − 8, find x-intercepts by setting H(x) = 0 and solving for x, then y-intercept by x = 0.
Inequalities (problems 8–10):
- Solve inequalities involving absolute values by splitting into two cases and solving for x.
- Example forms: 6 < |x − 15|; 2 − 3|x − 12| ≥ −4; |47 − 3|x| − 5 > 10, etc.
Conceptual questions (problem 11–13):
- Determine whether a set of expressions represents a solution set to an inequality or a function of x, and interpret intercepts, ranges, and solution sets.
Absolute value inequalities often yield two intervals or unions of intervals depending on the bounds.

Page 10–11: Composition of Functions (f ∘ g, g ∘ f, etc.)

The composition rule: (f ∘ g)(x) = f(g(x)); (g ∘ f)(x) = g(f(x)). Always evaluate the inside function first, then the outside function.
Example (clear from transcript):
- Let f(x) = x^2 + x and g(x) = 2x + 1.
- (a) f(g(3)):
- g(3) = 2(3) + 1 = 7;
- f(7) = 7^2 + 7 = 56.
- (b) g(f(3)):
- f(3) = 3^2 + 3 = 12;
- g(12) = 2(12) + 1 = 25.
Example: U(t) = 3t^2 − 8 and V(t) = −6t + 7
- (a) U(V(1)) = U(1) = 3(1)^2 − 8 = −5.
- (b) U(V(−1/2)) = V(−1/2) = −6(−1/2) + 7 = 3 + 7 = 10; U(10) = 3(10)^2 − 8 = 292.
Example: p = 2q^2 − 5 and q = 10 − 3r
- (a) If r = 2, q = 10 − 3(2) = 4; p = 2(4^2) − 5 = 27.
- (b) If p = 67, solve 67 = 2(10 − 3r)^2 − 5 ⇒ 72 = 2(10 − 3r)^2 ⇒ (10 − 3r)^2 = 36 ⇒ 10 − 3r = ±6 ⇒ r ∈ {4/3, 16/3}.
More compositions (6–8):
- Example: f(x) = √(7 + x), g(x) = 9x^2 − 7
- (a) f(g(x)) = √(7 + (9x^2 − 7)) = √(9x^2) = 3|x| (domain restrictions apply from the square root and inside expression).
- (b) g(f(x)) = 9(√(7 + x))^2 − 7 = 9(7 + x) − 7 = 63 + 9x − 7 = 56 + 9x.
Additional pattern questions (6–7):
- When computing nested functions, ensure the inner output remains within the domain of the outer function.
Practical exercise: build intuition by testing multiple inside-out evaluations and verifying results algebraically.

Page 12–15: Function Transformations, Inverses, and Graphical Transformations

12–13: Function Transformations (Translations, Reflections, Scalings)

Common transformations of a parent function f(x):
- f(x − a) shifts graph to the right by a units.
- f(x) − b shifts graph down by b units.
- f(x − a) + b shifts right by a and up by b.
- f(−x) reflects about the y-axis.
- −f(x) reflects about the x-axis.
- −f(−x) reflection about both axes (rotate across both axes).
- c·f(x) scales vertically by factor c (c > 0 stretches; 0 < c < 1 compresses).
- f(cx) scales horizontally by factor 1/c (if c > 1, horizontal compression; 0 < c < 1, horizontal stretch).
Example: 9f(x) − 9 scales vertically by 9 and shifts down by 9.
Example: (1/2)f(x) compresses vertically by a factor of 2.
Example: f(3 − x) is a horizontal reflection about the line x = 1. (Note: in simple terms, f(x+1) shifts left 1; f(−x) reflects about y-axis; combine with inside/outside factors accordingly.)
Table-based practice (x values and f(x) values) reinforces understanding of how input/output pairs change with transformations.
Graph interpretation tasks (15–18) involve identifying the parent function and the transformation from given graphs.

14–16: Inverse Functions and Invertibility

Key idea: A function is invertible if it is one-to-one (no two inputs map to the same output). If f is invertible, there exists a function f⁻¹ such that f(f⁻¹(y)) = y and f⁻¹(f(x)) = x.
How to test and find inverses: swap x and y and solve for y; check that the resulting relation defines y as a function of x.
Some functions are not invertible unless their domain is restricted to make them one-to-one (e.g., f(x) = x^2 is not one-to-one on all reals, but is invertible on [0, ∞) or (−∞, 0]).
Examples from transcript (typical patterns you should practice):
- Linear functions are invertible with inverse f⁻¹(x) = (x − b)/m when m ≠ 0.
- f(x) = x^3 + 1 is invertible on all reals (since x^3 is strictly increasing).
- f(x) = 7x + 3 has inverse f⁻¹(y) = (y − 3)/7.
- f(x) = x^2 is not invertible on R; it becomes invertible if domain is restricted to x ≥ 0 or x ≤ 0.
Worked examples (from transcript):
- Example: f(x) = x^3 + 1; g(x) = −3x + 8. Show that f and g can be inverses under appropriate domain restrictions, or show how to compute inverses by swapping and solving.
- In several items, the process is to swap x and y and solve for y to obtain f⁻¹(x); then check composition to confirm invertibility.
Important takeaway: If two functions are inverses, their compositions in either order yield the identity: f(g(x)) = x and g(f(x)) = x on the relevant domain.

Page 12–14: Practice with Specific Inverse Problems and Graphs

Problems 5–7 (inversions, composition):
- Identify whether two given functions are mutual inverses by examining f ∘ g or g ∘ f and by checking whether the compositions reduce to x (over the domain in question).
Problems 8–10: More inverse-testing and composition construction
- Use the identity f(g(x)) = x to verify inverses; if not, adjust domain to enforce one-to-one behavior.
Problems 11–12: Inverse pairs; determine existence of inverses for given f and g, and explain why or why not.
Problems 13–16: Graphing a function and its inverse; recognizing that the inverse graph is the reflection of the original graph across the line y = x.
Practical tip: When graphing inverses, reflect across the y = x line; check by swapping coordinates in point pairs.

Page 15: Consolidated Inverse Exercise Answers (Representative Examples)

Several items illustrate that some functions are invertible and some are not, depending on whether they are one-to-one across their domain.
Representative conclusions:
- A cubic function f(x) = x^3 + 1 is invertible on all reals (one-to-one).
- A quadratic function f(x) = ax^2 + bx + c is not invertible on all reals unless a restriction is placed on the domain (e.g., x ≥ 0 or x ≤ 0).
- Graphical inverses correspond to reflecting the original graph across the line y = x.
Note: The transcript includes several structured solutions with steps like “swap x and y” and solving for y; the pattern is consistent with standard inverse-finding procedures.

Quick Reference: Common Formulas and Notation (LaTeX)

Functions and composition
- $f(x)=ax+b, ext{ slope } a, ext{ intercept } b.$
- $(f\circ g)(x) = f(g(x)).$
- Inverse (if exists): solve for x in terms of y after swapping x and y.
Domain/range relations
- For $y=\sqrt{g(x)}$ , require $g(x)\ge 0$ and yield $\text{Domain} \subseteq {x: g(x)\ge 0}.$
- For $y=|h(x)|$ , range is $[0,\infty)$ unless constraints modify it.
Transformations (notation quick guide)
- Horizontal shift: $f(x-a)$ shifts right by $a$ .
- Vertical shift: $f(x) + b$ shifts up by $b$ .
- Reflection about x-axis: $-f(x)$ .
- Reflection about y-axis: $f(-x)$ .
- Vertical stretch/compression: $c f(x)$ with c>1 stretch, 0<c<1 compress.
- Horizontal stretch/compression: $f(cx)$ with c>1 compress horizontally, 0<c<1 stretch.
Intercepts (quick rules)
- x-intercept(s): set y = 0, solve for x.
- y-intercept(s): set x = 0, solve for y.

Study Tips and Connections

Build fluency with identifying domains and ranges from both function definitions and graphs; practice both directions (given domain, find range; given range, find domain).
For increasing/decreasing questions, always consider the slope (or derivative in calc contexts) and the interval of interest.
When modeling with real data (linear modeling), interpret slope as rate of change and discuss units; always consider possible sources of error when the model deviates from data.
Inverse problems: always test invertibility before assuming an inverse exists; use the horizontal/vertical line/tests as needed.
Transformations are best understood by thinking about how horizontal and vertical shifts, reflections, and scalings move the graph of the parent function. Practice with at least one example of each kind to become fluent.

If you want, I can convert these notes into a printable PDF with the same structure and LaTeX-ready formulas, using the notes above as a clean, organized study sheet.