Mr.Elliot's Math Cards :(

What is a Function?

Let X and Y be two non empty sets. A function from X into Y is relation that associates with each element of X exactly one element of Y

Complex Numbers

Numbers with the imaginary ‘i’

Real Numbers

All numbers that aren’t complex

Rational Numbers

Fractions, where the denominators != 0

How are lists made?

Using {}, Example: {7,9} or {(3,4),(-5,1)}

What makes a function?

One X CAN NOT have multiple Y’s, but multiple X’s can share one Y

{(9,4),(2,4),(1,-9)(9,6)} = Function?

No, the x, 9, has multiple Y’s, so this IS NOT a function

Not every function establishes y as a function of x

Example: y^2 = 4, y = 2 or y = -2 so this equation does not determine y as a function of x

Domain

For a function defined by a formula, the domain is assumed to be the set of all real numbers for which the function is defined and real.

\
This means:

\-do not divide by 0

\-do not take the root of a negative

How do you get the domain of 1/x+5?

because the denominator has X, set it to != 0. (i.e x+5 != 0), then remove the result from the domain. In this example the domain is (-infinity, -5) U (-5, infinity)

how do you get the domain of sqrt(x-5)

because there is a radical, set it ‘

What is the domain of sqrt(x+3)/x+1

x + 3 < 0 = x < -3

x + 1 = 0 = x != 1

so, the domain is \[-3,-1) U (-1, infinity)

what is an acceptable answer?

I used my calculator

When is a function even?

When it is symetric about the y axis

when is a function odd?

When it is symetric about the origin

Remember to us ‘U’ when talking about two different increasing sections

constant function

height is constant

f(x) = C

(-1,c) , (0,c) , (1,c)

identity function

identity of x doesn’t change

f(x) = x

(-1,-1) , (0,0) , (1,1)

absolute value function

measures distance from 0

f(x) = |x|

(-1,-1) , (0,0) , (1,1)

square function

like absolute value, different shape

f(x) = x^2

(-1,-1) , (0,0) , (1,1)

square root function

begins at the origin

f(x) = sqrt(x)

(0,0) , (1,1) , (4,2)

cube function

(negative \* *negative)* \* negative = negative

f(x) = x^3

(-1,-1) , (0,0) , (1,1)

cube root function

domin is (-infinity,infinity), passes V.L.T,

cube function sideways

f(x) = ^3sqrt(x)

(-1,-1) , (0,0) , (1,1)

reciprocal function

split into two, has 4 points

f(x) = 1/x

(1,1) , (2, 1/2) , (-2, -1/2) , (-1,-1)

Greatest integer function/ floor function

f(x) = \[\[x\]\] or f(x) = \[x\]

At an integer and NOT to the RIGHT of X

\[\[2\]\] = 2

\[2\] = 2

\[\[2.1\]\] = 2

\[2.3\] = 2

\[3\] = 3

\[-2\] = -2

\[-2.1\] = -3

PieceWise

if the rule applies, use it.

left

f(x+c)

right

f(x-c)

up

f(x)+c

down

f(x)-c

compressed by c (Vert.)

cf(x), c

stretched by c(Vert.)

cf(x), c>1

reflect over X

\-f(x)

Reflect over Y

f(-x)

stretched by C (horizontal)

f(cx), c

compressed by c (horizontal)

f(cx), c>1

What is the basic function?

f(x) = x^2

Order of transformations?

use pendas, and remember in -2f(x), the - and 2 are seperate steps

stretch in the negative?

vert stretch by 2, go down by 2