Functions, Graphs & Logarithms

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medium importance25 formulas⚡ 15 shortcuts5 subtopics

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Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.

Functions, domain & range

Domain: denominator

x \ne \text{(zeros of the denominator)}

Reject every input that makes the bottom zero.

Domain: square root

\sqrt{A} \text{ needs } A \ge 0

The inside of a root stays on the non-negative side.

Range of a quadratic

a(x-h)^2 + k \Rightarrow \text{range} = [k, \infty) \text{ if } a > 0

Complete the square; $k$ is the floor or the ceiling.

Vertex input

x = -\frac{b}{2a}

Substitute back to get the minimum or maximum value.

Even / odd test

f(-x) = f(x) \text{ (even)}; \quad f(-x) = -f(x) \text{ (odd)}

Replace $x$ by $-x$ and simplify fully before deciding.

Composite & inverse functions

Composite

(f \circ g)(x) = f(g(x))

$g$ acts first; work inside out.

Inverse check

f(f^{-1}(x)) = f^{-1}(f(x)) = x

Composition both ways must return $x$.

Inverse of (ax+b)/(cx+d)

f^{-1}(x) = \frac{b - dx}{cx - a}

From swap-and-solve; memorise for speed.

Sum-type equation

f(x+y) = f(x) + f(y),\ f(1) = k \Rightarrow f(x) = kx

Each unit step adds $k$.

Product-type equation

f(x+y) = f(x)f(y),\ f(1) = k \Rightarrow f(x) = k^x

Each unit step multiplies by $k$.

Graphs & transformations

Vertical shift

y = f(x) + k

$k > 0$ lifts the picture, $k < 0$ lowers it.

Horizontal shift

y = f(x - a)

Moves right by $a$; $f(x+a)$ moves left.

Reflections

y = -f(x), \quad y = f(-x)

First flips over the $x$-axis, second over the $y$-axis.

Modulus graph

y = |f(x)|

The part below the axis is reflected up.

Vertex of a parabola

x = -\frac{b}{2a}, \quad \text{value} = f\left(-\frac{b}{2a}\right)

Minimum if $a > 0$, maximum if $a < 0$.

Logarithms

Definition

\log_a b = x \Leftrightarrow a^x = b

Base $a > 0$, $a \ne 1$; argument $b > 0$.

Product / quotient laws

\log_a(mn) = \log_a m + \log_a n, \quad \log_a\frac{m}{n} = \log_a m - \log_a n

Multiplication becomes addition; division becomes subtraction.

Power law

\log_a m^p = p \log_a m

The exponent comes out front.

Change of base

\log_a b = \frac{\log b}{\log a}

Any common base works; base 10 is standard.

Special values

\log_a 1 = 0, \quad \log_a a = 1, \quad a^{\log_a b} = b

Log of 1 is 0; log of the base is 1; base and log cancel.

Modulus & equations

Modulus equation

|x - a| = d \Rightarrow x = a \pm d

Two points at distance $d$ from $a$.

Inside the band

|x| < a \Rightarrow -a < x < a

One interval; needs $a > 0$.

Outside the band

|x| > a \Rightarrow x < -a \text{ or } x > a

Two arms; the sign '>' splits.

Distance sum

\min\big(|x-a| + |x-b|\big) = |a - b|

Every $x$ between $a$ and $b$ achieves it.

Exponential match

a^{f(x)} = a^{g(x)} \Rightarrow f(x) = g(x)

Write both sides with the same base first.