NDA · Chapter

Functions

The concepts NDA needs, one connected idea at a time.

0 / 11

concepts mastered

Functions0/5
Basic functions0/3
Combining functions0/3

Ready for you

Up nextWhat makes a relation a functionEvery element of the domain has exactly one image; a vertical line meets the graph at most onceStart

Whole chapter

How the ideas build on each other. Each one opens once you've mastered what it needs.

Functions0 / 5

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 $x$ ,

Basic functions0 / 3

Modulus and signum $|x| = x$ for $x \ge 0$ and $-x$

Combining functions0 / 3

Algebra of functions $(f + g)(x) = f(x) + g(x)$ ;

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

Functional equationsFind

f

from a rule it obeys: put in clever values such as

x = 0

or

x \to \frac{1}{x}

, or recognise

f(x + y) = f(x) + f(y)

and its relativesLocked

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

;

\{x\} = x - [x]

Locked

Piecewise defined functionsFind which piece the input falls in first, including which piece owns the boundaryLocked

\frac{f}{g}

needs

g(x) \ne 0

as well as both domains

Locked

Composition of functions

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

:

g

acts first;

f \circ g \ne g \circ f

in generalLocked

Inverse functionsOnly a one-one onto

f

has

f^{-1}

: write

y = f(x)

, solve for

x

;

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

Locked