MH Board · Chapter

Functions

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

;

\log_a x = y \iff a^y = x

brackets and all;

f(x + 1)

is a new expression, not

f(x) + 1

Locked

Domain and rangeExclude zero denominators, negatives under even roots and non-positive log argumentsLocked

One-one, onto and how many functionsOne-one: different inputs, different outputs; onto: every co-domain element is hit;

n^m

functions from

m

to

n

elementsLocked

Modulus and signum $|x| = x$ for $x \ge 0$ and $-x$ for $x < 0$ ; $\operatorname{sgn}x = \frac{|x|}{x}$ , with $\operatorname{sgn}0 = 0$ Locked
Greatest integer and fractional part $[x]$ is the integer at or below $x$ : $[-2.3] = -3$ , not $-2$ ;
Logarithm rules $\log mn = \log m + \log n$ , $\log m^k = k\log m$ ,
Piecewise defined functionsFind which piece the input falls in first, including which piece owns the boundaryLocked

Algebra of functions $(f + g)(x) = f(x) + g(x)$ ; $\frac{f}{g}$ needs $g(x) \ne 0$ as well as both domainsLocked
Composition of functions $(f \circ g)(x) = f(g(x))$ : $g$ acts first; $f \circ g \ne g \circ f$
Inverse functionsOnly a one-one onto $f$ has $f^{-1}$ : write $y = f(x)$ ,

\{x\} = x - [x]

Locked

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

;

\log_a x = y \iff a^y = x

Start

in general

Locked

solve for

x

;

f^{-1}(x) \ne \frac{1}{f(x)}

Locked