Special Functions: Chapter Review

Vocabulary and notation

Radical function

A function with the variable under a root, as in $\sqrt{x}$ or $\sqrt[3]{x + 4}$. Its index names the root ($2$ when unwritten); the radicand sits under it.

Principal square root

$\sqrt{x}$ names the root that is not negative, so $\sqrt{16} = 4$ alone, never $\pm 4$.

Extraneous solution

A candidate satisfying the equation you produced but not the original. Squaring, or clearing a variable denominator, can manufacture one.

Absolute value $\lvert x \rvert$

The distance from $x$ to $0$ on the number line, so $\lvert -5 \rvert = \lvert 5 \rvert = 5$.

Vertex

The corner of a V graph: lowest point when it opens up, highest when it opens down.

Floor $\lfloor x \rfloor$ and ceiling $\lceil x \rceil$

The greatest integer at or below $x$, and the least at or above it: the nearest integer LEFT, and RIGHT. Both are staircases, domain all reals, range the integers.

Step function

A piecewise function with constant pieces, graphing as flat segments. The floor, the ceiling, and tiered shipping are the examples.

Rational function

A ratio of two polynomials, $f(x) = \dfrac{p(x)}{q(x)}$, with $q$ not the zero polynomial.

Asymptote

A straight line the graph approaches ever more closely outward. A guide-line, not part of the graph.

Piecewise function

A function given by two or more formulas, its pieces, each with a condition naming the inputs it governs.

Boundary

An input where one piece's condition ends and the next begins. The inequality signs decide which piece owns it.

Formulas and theorems

• Domain of a radical

  Even index: solve $\text{radicand} \ge 0$. Odd index: every real is allowed. A root is a fractional power: $\sqrt[n]{x} = x^{1/n}$.

  Use when Even $n = 2, 4, 6, \dots$: a real even power is never negative. Odd $n = 3, 5, \dots$: $\sqrt[3]{-8} = -2$ is genuine.

• The three parent graphs

  $y = \sqrt{x}$: from the corner $(0, 0)$, climbing ever more slowly. $y = \lvert x \rvert$: a V, vertex $(0, 0)$, branch slopes $-1$ and $+1$. $y = \dfrac{1}{x}$: a hyperbola, two branches never joined, asymptotes $x = 0$ and $y = 0$.

  Text description

  Three panels: a square-root curve, an absolute-value V, and the reciprocal's two branches.

  Use when Ranges: $y \ge 0$ for the first two. The square-root curve is the upper half of the sideways parabola $y^2 = x$.

• Shifted parent forms

  $$\begin{gathered} y = a\sqrt{x - h} + k \\ y = a\lvert x - h \rvert + k \\ y = \frac{a}{x - h} + k \end{gathered}$$

  Use when $a \neq 0$. The anchor moves to $(h, k)$: corner, vertex, or where $x = h$ meets $y = k$. Domains: $x \ge h$, all reals, all reals but $h$. Radical and V ranges: $y \ge k$ if $a > 0$, else $y \le k$. Larger $\lvert a \rvert$ narrows a V; $a < 0$ flips across $y = k$.

  e.g. $y = -2\sqrt{x}$ keeps domain $x \ge 0$ but flips the range to $y \le 0$.

• Piecewise rule for absolute value

  $$\lvert x \rvert = \begin{cases} x & \text{if } x \ge 0, \\ -x & \text{if } x < 0. \end{cases}$$

  Use when Every real $x$; $\lvert x \rvert \ge 0$ always, equality only at $x = 0$. Split a shifted bar on the sign of its inside: $\lvert x - 3 \rvert$ is $x - 3$ for $x \ge 3$, else $3 - x$.

• Absolute value equations and inequalities

  $\lvert X \rvert = c$: two cases $X = c$ or $X = -c$ when $c > 0$; one case $X = 0$ when $c = 0$; no solution when $c < 0$. $\lvert X \rvert < c$: one band, $-c < X < c$. $\lvert X \rvert > c$: two rays, $X < -c$ or $X > c$.

  Use when Isolate the bars first; $X$ may be any expression. The inequality forms need $c > 0$; $\le$ and $\ge$ read the same, endpoints included.

• Floor and ceiling, pinned by inequalities

  $$\begin{gathered} \lfloor x \rfloor \le x < \lfloor x \rfloor + 1 \\ \lceil x \rceil - 1 < x \le \lceil x \rceil \end{gathered}$$

  Text description

  A staircase of unit steps, each closed at its left end and open at its right.

  Use when Every real $x$; the integer satisfying either is unique. The brackets agree on the integers, where $\lfloor x \rfloor = \lceil x \rceil = x$, and differ by exactly $1$ elsewhere; $\lceil x \rceil = -\lfloor -x \rfloor$ turns either into the other. And $\lfloor x + \tfrac{1}{2} \rfloor$ rounds to the nearest integer, ties up.

  e.g. $\lceil x \rceil = -1$ holds for exactly the inputs $-2 < x \le -1$.

• Fractional part

  $$\{x\} = x - \lfloor x \rfloor, \qquad 0 \le \{x\} < 1$$

  Use when Every real $x$, negatives included, and $0$ exactly on the integers. For a negative non-integer it is NOT the digits after the decimal point.

  e.g. $\{-1.25\} = -1.25 - (-2) = 0.75$.

• Vertical asymptote or hole

  Cancel as far as it will go, then read the REDUCED denominator: if $x - a$ still divides it, $x = a$ is a vertical asymptote; if not, a hole at the reduced form's height.

  Use when Either way $x = a$ must be a zero of the ORIGINAL denominator; only complete factoring separates the cases.

  e.g. $\dfrac{(x - 3)(x + 2)}{(x - 3)(x + 3)}$: hole at $\left(3, \tfrac{5}{6}\right)$, asymptote $x = -3$.

• Horizontal asymptote by degrees

  $n < m$: $y = 0$. $n = m$: $y = \dfrac{a}{b}$, the ratio of leading coefficients. $n > m$: none.

  Use when $n$, $m$ are the degrees and $a$, $b$ the leading coefficients of top and bottom. When $n = m + 1$ a slant line takes the asymptote's place, found by dividing.

  e.g. $\dfrac{3x + 2}{x^2 - 1}$ settles toward $y = 0$, while $\dfrac{x^3 + 1}{x^2 - 5}$ has none.

• Piecewise: partition, dots, and the jump test

  Conditions must give each input exactly one piece. Close the dot on the piece that includes a boundary, open it on the one that excludes it, then evaluate BOTH formulas there: equal heights connect, different heights jump.

  Use when Pair $x \le b$ with $x > b$, or $x < b$ with $x \ge b$; pairing $\le$ with $\ge$ claims $b$ twice, $<$ with $>$ not at all. A connected boundary can still be a corner, as at the vertex of $\lvert x \rvert$.

Problem types, step by step

Exam traps

• Trap Rounding a negative toward zero, so $\lfloor -2.3 \rfloor = -2$.

  Fix The floor goes DOWN, toward negative infinity: $\lfloor -2.3 \rfloor = -3$, while $\lceil -2.3 \rceil = -2$. Dropping the digits is truncation, a different move.

• Trap Reading the shift straight off the sign inside, so $y = \sqrt{x + 3} - 5$ looks like a shift right $3$.

  Fix Match $x - h$: $x + 3$ is $x - (-3)$, so $h = -3$, the graph goes LEFT $3$ and down $5$, anchor $(-3, -5)$. The same reversal hits $\lvert x + 3 \rvert$ and $\dfrac{1}{x + 3}$.

• Trap Splitting an absolute value equation before the bars are alone: $2\lvert x + 1 \rvert - 3 = 5$ giving $x + 1 = \pm 5$.

  Fix That solves a different equation. Reduce to $\lvert x + 1 \rvert = 4$ first, giving $x = 3$ or $x = -5$, not $4$ or $-6$.

• Trap Distributing bars across a sum: $\lvert a + b \rvert = \lvert a \rvert + \lvert b \rvert$.

  Fix $\lvert 3 + (-8) \rvert = 5$, while $\lvert 3 \rvert + \lvert -8 \rvert = 11$. They agree only when $a$ and $b$ share a sign or one is $0$.

• Trap Squeezing a "greater than" answer into one chain, as $-7 \ge x \ge 1$.

  Fix Two rays are two pieces: write $x \le -7$ or $x \ge 1$. A chain claims $x$ is below $-7$ and above $1$ at once.

• Trap Treating a cancelled factor's input as an ordinary point of the graph.

  Fix Cancelling never restores an input. After $x - 3$ cancels, $x = 3$ stays barred: the reduced curve, punched out there.

• Trap Reading a horizontal asymptote off the constant terms, or assuming a curve can never cross one.

  Fix Leading terms decide: $\dfrac{4x^2 - x}{2x^2 + 7}$ gives $y = 2$, not $\tfrac{0}{7}$, and a higher degree on top leaves none. Crossing is allowed: only a vertical asymptote is off limits, where the function has no value.

• Trap Deciding which candidate is extraneous by size, or assuming exactly one always fails.

  Fix Only substitution decides: in $\sqrt{7x - 5} = x + 1$ both $x = 2$ and $x = 3$ survive, while a rational equation can lose every candidate.

• Trap Joining the two ends of a jump with a vertical segment.

  Fix A jump is a gap: an open dot and a closed dot at different heights, nothing between, because the function skips those values.