MHT-CET · Chapter

Sets, Relations and Functions

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

;

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

subsets, counting

\varnothing

and itself

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Interval notationSquare bracket: the end is in; round bracket: it isn't; $\infty$ always gets a round oneLocked
Operations on intervalsUnion, intersection, difference and complement of intervals, read off the number lineLocked
Solving inequalities as intervalsLinear: flip the sign when dividing by a negative; quadratic and rational: test the sign between critical pointsLocked

Ordered pairs and the Cartesian product $(a, b) = (c, d)$ only when $a = c$ and $b = d$ ; $n(A \times B) = n(A)\,n(B)$ Locked
Relations: domain, co-domain and rangeA relation is a subset of $A \times B$ ; its domain is the first entries used, its range the secondLocked

What makes a relation a functionEvery element of the domain has exactly one image; a vertical line meets the graph at most onceStart
Values of a function $f(a)$ : put $a$ for every $x$ , 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]

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

;

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

Start

in general

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solve for

x

;

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

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