Solve set: functions, exp/log, trig and Euler, series, polynomials, inequalities

Overview

Module	S-M00 · solve · pen and paper, checked by SymPy · Pass 2 · 4 to 6 h
You write	solve/S-M00.toml in your repo: 60 answers in ASCII math (ol start S-M00 writes the template)
Problems	course/solve/S-M00/problems.md, reproduced in section 4
Checked by	ol check S-M00: each answer is compared with a hidden key by SymPy (symbolically, then numerically at seeded points); a part passes at 80%
Needs	nothing. Each part points to the chapter section that teaches it
Used by	the four build modules of this topic (M00.1 to M00.4) as their pen-and-paper check; parts 1 and 6 are M00.5 (functions, inverses, monotonicity, inequalities), which M01.1 and M05.2 assume
Milestone	MS-P2 (the Pass 2 gate: every math module and solve part of the pass)
Optional depth	OpenStax, Precalculus 2e (free), ch. 1 (functions), 2 to 3 (linear and polynomial functions, inequalities), 6 (exponentials and logs), 5 to 8 (trigonometry, complex numbers), 11 (sequences and series)

Key Takeaways

• A function assigns exactly one output to each input in its domain; its inverse exists exactly when no two inputs share an output, and undoes it: $f^{-1}(f(x)) = x$.
• A strictly monotonic function (always increasing, or always decreasing) is one-to-one, so it has an inverse; that is why $\exp$, $\ln$, the sigmoid, and softplus can all be inverted.
• Solving an inequality is solving the matching equation and testing the sign between its solutions; multiplying or dividing by a negative number, or applying a decreasing function, flips the direction.
• The other four parts are the pen-and-paper side of M00.1 to M00.4: if a problem in part 2 to 5 stops you, read the beat 2 section of that module’s chapter it names.

How to work this chapter

ol start S-M00              # writes solve/S-M00.toml with an empty answer per question
$EDITOR solve/S-M00.toml    # answer = "(x + 7)/3", one line per question
ol check S-M00              # per-question pass/fail; feedback never shows the expected answer

Answers are ASCII math: x^2, sqrt(x), log(x) (natural log), E, pi, I, oo; intervals like (-oo, 2) U [3, 5); sets {1, 2}; vectors [1, 2]. A wrong answer reports where it differs (q4 FAIL: differs at x=1.30), not what it should be.

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1. Why now

Pass 2 turns the math under your tracer system into code: logarithms that measure information (M00.1), rotations that encode position (M00.2), ladders of frequencies (M00.3), and polynomial evaluation that kernels run (M00.4), and after them calculus, linear algebra, probability, and optimization. Each of those chapters assumes you can manipulate the expressions fluently by hand: invert a function, combine logarithms, use an angle-addition formula, sum a geometric series, factor a polynomial, and solve an inequality. A bug in a derivation becomes a bug in code that tests catch late and explain poorly. This set checks the hand skills first, with a checker that accepts any correct form of an answer. Parts 1 and 6 also teach M00.5, functions, inverses, monotonicity, and inequalities, which has no build module of its own because none of it becomes code; M01.1 (limits and derivatives) and M05.2 build on it directly.

2. Principles

Symbol	Meaning	Type / shape
$f: A \to B$	a function from the set $A$ (its domain) to the set $B$
$f(x)$	the output of $f$ at the input $x$
$\operatorname{dom} f$	the domain: the inputs where $f$ is defined	a set or interval
$\operatorname{ran} f$	the range: the outputs $f$ actually takes	a set or interval
$f \circ g$	composition, $(f \circ g)(x) = f(g(x))$
$f^{-1}$	the inverse function: $f^{-1}(y) = x$ exactly when $f(x) = y$
$(a, b)$, $[a, b]$	the open interval $a < x < b$ and the closed interval $a \le x \le b$
$\cup$	union of sets, written U in answers
$\sigma(z)$	the sigmoid $1/(1 + e^{-z})$
$\lvert x\rvert$	absolute value: $x$ if $x \ge 0$, $-x$ otherwise

2.1 Functions, domains, ranges, composition

A function $f$ assigns to each input $x$ in its domain exactly one output $f(x)$. A formula alone leaves the domain implicit: it is every real $x$ where the formula makes sense. $\ln(4 - x^2)$ needs $4 - x^2 > 0$, so its domain is $(-2, 2)$; $1/(x - 3)$ needs $x \ne 3$. The range is the set of outputs that actually occur: the sigmoid $\sigma(z) = 1/(1 + e^{-z})$ takes every value strictly between 0 and 1 and never reaches either, so its range is $(0, 1)$. Composition chains functions: $(f \circ g)(x) = f(g(x))$ applies $g$ first. Order matters: with $f(x) = x^2$ and $g(x) = x + 1$, $f(g(x)) = (x + 1)^2$ but $g(f(x)) = x^2 + 1$.

2.2 Inverses and monotonicity

$f$ is one-to-one when different inputs give different outputs. Then every output $y$ in the range comes from exactly one $x$, and the inverse $f^{-1}(y) = x$ undoes $f$: $f^{-1}(f(x)) = x$ and $f(f^{-1}(y)) = y$. To find it, write $y = f(x)$ and solve for $x$; the graph of $f^{-1}$ is the graph of $f$ reflected across the line $y = x$.

A function is strictly increasing when $x_1 < x_2$ implies $f(x_1) < f(x_2)$ (strictly decreasing: $f(x_1) > f(x_2)$). Either kind, called strictly monotonic, is one-to-one, because different inputs are ordered and so are their outputs. $e^x$, $\ln x$, $x^3 + x$, $\sigma$, and softplus $\ln(1 + e^x)$ are strictly increasing; $x^2$ is not monotonic on all of $\mathbb{R}$ ($(-2)^2 = 2^2$), so it has an inverse only on a half-line, where $\sqrt{\ }$ is it. M01.1 gives the practical test: a function whose derivative is positive everywhere is strictly increasing.

2.3 Inequalities

An inequality is solved the way an equation is, with one extra rule: multiplying or dividing both sides by a negative number reverses the direction ($-2x < 6$ means $x > -3$), and so does applying a strictly decreasing function. Applying a strictly increasing function ($e^x$, $\ln$, $\sqrt{\ }$ on non-negative numbers) keeps it, which is how $e^x \le 2$ becomes $x \le \ln 2$.

For a polynomial or a quotient of polynomials, find where it is zero or undefined, cut the line there, and test the sign on each piece. $x^2 - 5x + 6 = (x - 2)(x - 3)$ is zero at 2 and 3, positive outside and negative between, so $x^2 - 5x + 6 > 0$ on $(-\infty, 2) \cup (3, \infty)$. An endpoint where the expression is zero belongs to the answer for $\ge$ and $\le$, never one where it is undefined. An absolute value is a distance: $|x - 3| < 2$ means “within 2 of 3”, the interval $(1, 5)$. Two inequalities appear again in optimization and information theory: $x + 1/x \ge 2$ for $x > 0$ (with equality at $x = 1$; multiply by $x$ and get $(x - 1)^2 \ge 0$), and $\ln x \le x - 1$ (equality only at $x = 1$), the inequality behind Gibbs’ inequality in M11.1.

2.4 Where the other parts are taught

Part	Questions	Taught in
2. exponents, logarithms, units	q11 to q22	M00.1 sections 2.1 to 2.4
3. trigonometry, complex numbers, Euler	q23 to q34	M00.2 sections 2.1 to 2.6
4. sequences and geometric series	q35 to q44	M00.3 sections 2.1 to 2.5
5. polynomials	q45 to q52	M00.4 sections 2.1 to 2.5

3. Worked example by hand

A worked solution of a sibling of q5 and q6: the inverse and range of $f(x) = 2/(1 + e^{-x})$.

1. Range. $e^{-x}$ takes every value in $(0, \infty)$, so $1 + e^{-x}$ takes every value in $(1, \infty)$ and $2/(1 + e^{-x})$ every value in $(0, 2)$, approaching 2 as $x \to \infty$ and 0 as $x \to -\infty$ without reaching either. Range: (0, 2).
2. Monotonic, so invertible. As $x$ grows, $e^{-x}$ shrinks, the denominator shrinks, and $f(x)$ grows: $f$ is strictly increasing, hence one-to-one on $\mathbb{R}$.
3. Solve $y = f(x)$ for $x$, for $0 < y < 2$: $y(1 + e^{-x}) = 2 \;\Rightarrow\; e^{-x} = \frac{2}{y} - 1 = \frac{2 - y}{y} \;\Rightarrow\; -x = \ln\frac{2 - y}{y} \;\Rightarrow\; x = \ln\frac{y}{2 - y}.$
4. Check. $f(0) = 2/(1 + 1) = 1$, and $f^{-1}(1) = \ln(1/1) = 0$. Correct.
5. Answer, as you would write it in solve/S-M00.toml (in the variable of the question): answer = "log(y/(2 - y))". The checker also accepts log(y) - log(2 - y) or -log(2/y - 1), because it tests equality, not spelling: symbolically first, then at 32 seeded points of the domain the key declares.

And one inequality, a sibling of q56: $\dfrac{x + 1}{x - 4} \le 0$. The numerator is zero at $-1$ and the denominator at $4$. Signs: for $x < -1$ both factors are negative and the quotient positive; for $-1 < x < 4$ the numerator is positive and the denominator negative, so the quotient is negative; for $x > 4$ it is positive. Zero at $x = -1$ is allowed by $\le$; $x = 4$ divides by zero. Answer: [-1, 4).

4. Problem set

Write each answer in solve/S-M00.toml as answer = "..." under its [qN] header. The tag after each question is the answer type.

Functions and inverses

q1. $f(x) = 3x - 7$. Give $f^{-1}(x)$. [expr in x]

q2. $f(x) = e^{2x + 1}$. Give $f^{-1}(x)$ for $x > 0$. [expr in x]

q3. $g(x) = \dfrac{2x + 1}{x - 3}$ for $x \ne 3$. Give $g^{-1}(x)$. [expr in x]

q4. $f(x) = x^2$ and $g(x) = x + 1$. Give $(f \circ g)(x) = f(g(x))$. [expr in x]

q5. The sigmoid is $\sigma(z) = 1 / (1 + e^{-z})$. Give its inverse, the logit $\sigma^{-1}(p)$, for $0 < p < 1$. [expr in p]

q6. Give the range of $\sigma$ (the set of values it takes over all real $z$). [interval]

q7. Give the domain of $h(x) = \ln(4 - x^2)$ (the real $x$ where it is defined). [interval]

q8. Is $f(x) = x^3 + x$ one-to-one on all of $\mathbb{R}$? (Hint: is it monotonic?) [bool]

q9. $f(x) = 2^x$ and $g(x) = \log_2 x$. Compute $f(g(16)) + g(f(3))$. [number]

q10. Softplus is $s(x) = \ln(1 + e^x)$. Give $s^{-1}(y)$ for $y > 0$. [expr in y]

Exponents, logarithms, and units of information

q11. $\log_2 8 + \log_2 4$. [number]

q12. $\log_8 32$. [number]

q13. Simplify $e^{3 \ln x}$ for $x > 0$. [expr in x]

q14. Solve $2^x = 1000$ for $x$. [number], decimal ok

q15. How many bits is one nat? [number], decimal ok

q16. A model’s mean negative log-likelihood is $2.0$ nats per token on a text of $1000$ tokens and $4000$ bytes. Give its bits per byte. [number], decimal ok

q17. A model’s perplexity is $e^{\text{mean NLL in nats}} = 8$ per token. Give its mean NLL in bits per token. [number]

q18. Write $a = \ln 2$ and $b = \ln 3$. Express $\ln 72$ in $a$ and $b$. [expr in a, b]

q19. Solve $\log_3 x + \log_3 (x - 8) = 2$. [number]

q20. Solve $4^x = 8$. [number]

q21. A quantity decays as $N(t) = N_0 e^{-kt}$ with $k > 0$. After what time is it half of $N_0$? [expr in k]

q22. How many decimal digits does $2^{100}$ have? (Use $\log_{10} 2 \approx 0.30103$.) [number]

Trigonometry, complex numbers, and Euler’s formula

q23. $\sin(\pi/6)$. [number]

q24. $\cos(2\pi/3)$. [number]

q25. Write $\cos a \cos b - \sin a \sin b$ as one trigonometric function of $a$ and $b$. [expr in a, b]

q26. Rotate the point $(1, 0)$ counterclockwise by $\pi/3$. Give the image. [vector]

q27. $R(\pi/2)\,R(\pi/3) = R(\theta)$ with $\theta \in [0, 2\pi)$, where $R(\theta)$ is the rotation by $\theta$. Give $\theta$. [number]

q28. $(1 + i)^8$. [number]

q29. $|3 + 4i|$. [number]

q30. Rotate $3 + 4i$ by the angle $t$ with $\cos t = 3/5$ and $\sin t = 4/5$: compute $(3 + 4i)(\cos t + i \sin t)$. [number]

q31. $e^{i\pi/2}$. [number]

q32. Express $\cos t$ using only exponentials $e^{it}$ and $e^{-it}$. [expr in t]

q33. For the unit complex numbers $u = e^{ia}$ and $v = e^{ib}$, simplify $\operatorname{Re}(u\,\bar{v})$, where $\bar{v}$ is the complex conjugate. [expr in a, b]

q34. A RoPE dimension pair rotates by $\omega p$ radians at position $p$, with $\omega = 0.01$. After how many positions does the pattern repeat (the period)? [number]

Sequences and geometric series

q35. $1 + \tfrac12 + \tfrac14 + \tfrac18 + \cdots$ (forever). [number]

q36. Give $\sum_{j=0}^{n-1} r^j$ in closed form for $r \ne 1$. [expr in r, n]

q37. The sum of the first 10 terms of $3, 6, 12, 24, \ldots$ [number]

q38. The RoPE ladder of a 128-dimensional head with base $10000$ is $\omega_i = 10000^{-2i/128}$ for $i = 0, \ldots, 63$. Give the ratio $\omega_{i+1}/\omega_i$. [number]

q39. $1 + 2 + \cdots + 100$. [number]

q40. An exponential moving average $m_t = \beta m_{t-1} + (1 - \beta) g_t$ starts at $m_0 = 0$ with $\beta = 0.9$ and sees $g_t = 1$ at every step. Give $m_3$. [number]

q41. In q40, $m_3 = (1 - \beta)(g_3 + \beta g_2 + \beta^2 g_1)$. Give the weight on the oldest gradient $g_1$. [number]

q42. $\lim_{n \to \infty} \dfrac{2n + 1}{n + 3}$. [number]

q43. Write $0.272727\ldots$ (27 repeating) as a fraction. [number]

q44. ALiBi with 4 heads uses the slopes $2^{-2}, 2^{-4}, 2^{-6}, 2^{-8}$. Give their sum. [number]

Polynomials

q45. Evaluate $p(x) = 2 - 3x + x^2 + 4x^3$ at $x = 2$ (Horner’s rule makes it three multiply-adds). [number]

q46. Divide $x^3 - 6x^2 + 11x - 6$ by $x - 1$. Give the quotient. [expr in x]

q47. Give all roots of $x^3 - 6x^2 + 11x - 6$. [set]

q48. Give the roots of $2x^2 + 3x - 2$. [set]

q49. Give the smaller root of $x^2 - 10^6 x + 1$ exactly (radicals allowed) or to 15 significant digits. [number]

q50. Without solving, give the product of the roots of $3x^2 - 7x + 2$. [number]

q51. How many real roots does $x^2 + x + 1$ have? [number]

q52. The Taylor polynomial of $e^x$ of degree 3 around 0 is $c_0 + c_1 x + c_2 x^2 + c_3 x^3$. Give $[c_0, c_1, c_2, c_3]$. [vector]

Inequalities

q53. Solve $|x - 3| < 2$. [interval]

q54. Solve $x^2 - 5x + 6 > 0$. [interval]

q55. The smallest integer $n$ with $2^n > 1000$ (the bits needed to number 1000 items). [number]

q56. Solve $\dfrac{x - 1}{x + 2} \ge 0$. [interval]

q57. For which probabilities $p \in (0, 1)$ is the surprisal $-\log_2 p$ greater than 3 bits? [interval]

q58. Solve $e^x \le 2$. [interval]

q59. The smallest value of $x + 1/x$ over $x > 0$. [number]

q60. Solve $\ln x < x - 1$ over $x > 0$. [interval]

5. Pitfalls

Pitfall	Where it bites later	Caught by
inverting the steps in the original order ($f^{-1}(x) = (x - 7)/3$ for $f(x) = 3x - 7$)	every inverse in M07.1 (inverse-CDF sampling)	q1, q3
composing in the wrong order: $g(f(x))$ instead of $f(g(x))$	the chain rule in M01.2 and backpropagation in L0.1	q4, q9
a closed endpoint where the expression is undefined	the domain of $\ln$ in every loss	q7, q56
$\log(a + b) = \log a + \log b$	rewriting losses in M11.1	q11, q18
one term too many, or the wrong weight, in a geometric sum	Adam’s bias correction (M02.2)	q37, q40, q41
decimals where an exact value is asked	none: an exact value is what the checker can prove equal	every exact number answer, with the message “give an exact value (no decimals)”

6. Where it’s used next

Direction	Module	How it uses this
Forward	M00.1	part 2 is the pen-and-paper side of its logarithms and units
Forward	M00.2	part 3: rotations, Euler’s formula, and the relative-angle identity of q33
Forward	M00.3	part 4: geometric sums, the RoPE ratio (q38), ALiBi slopes (q44), EMA weights (q40, q41)
Forward	M00.4	part 5: Horner (q45), factoring, Vieta, and the cancellation case (q49)
Forward	M01.1	limits (q42) and monotonicity (q8) start calculus
Forward	M05.2	functions, inverses, and bijections in discrete math
Forward	M07.1	inverse functions become inverse-CDF sampling