12 Basic functions

Identity Function

f(x)=x

Increasing, D infinite, R infinite, continuous, No extrema, No discontinuity, unbounded, odd function

Squaring Function

f(x)=x²

Increasing over (0, infinity), decreasing over (-infinity, 0), D infinite, R [0, infinity), never constant, Minimum extrema at (0, 0), below at y=0, continuous, even function

Cubing Function

f(x)=x³

domain infinite, range infinite, increasing, no extrema for parent

NATURAL log function

f(x)=lnx

Cannot go past x=0, asymptote there, root at (1,0)

RECIPROCAL function

asymptote at x=0 and y=0, odd,infinite discontinuity

find the vert asymptote at denominator=0 and find horiz asymptote using the weird rules

IF BOTH DEGREES ARE EQUAL, HORIZONTAL ASYMPTOTE IS AT THE TOP BIGGEST DEGREE DIVIDED BY BOTTOM BIGGEST DEGREE
IF TOP IS LESS THAN BOTTOM DEGREE, HORIZONTAL ASYMPTOTE AT y=0

IF TOP IS BIGGER BY 1, DO LONG DIVISION TO FIND OBLIQUE ASYMPTOTE

Exponential Function

f(x)=e^x
domain infinite, range [0, infinity), y-intercept at (0,1), increasing, no extrema, never constant, below at y=0, continuous, no symmetry

Absolute Value Function

f(x)=|x|
Range [0, infinity)

domain infinite

extrema at (0,0)

Below at y=0

Even function

Continous

Can use absolute value to make it zig zag, can be used for any function

Sine Function

f(x)=sinx

Domain infinite, range [-1, 1] (WILL BE TRANSFORMED)

y intercept at (0,0)

1 hill 1 peak every pi

Bounded by both

continous

ODD FUNC

Cosine Function

f(x)=cos(x)

Basically the same as sine func, domain same, range [-1, 1], except

y intercept at 1

Sine func but shifted 0.5 pi left
EVEN FUNC

Greatest Integer func

point at (0,0) aka y intercept

domain is infinite

range is (fancy bracket)…-3, -2, -1, 0, 1, 2, 3,…(fancy bracket)
special bracket means greatest integer less than or equal to x
Discontinuity: jump discontinuity at x E Z (DOESNT MATTER WHAT TRANSFORMATIONS ARE PERFORMED, IT IS STILL x E Z)
Unbounded

Roots: [0,1)

Logistic Function

f(x)=1/(1+e^(-x))
y intercept at (0, 1/2)

Range: (1, 0)

Horiz asymptotes at y=1, y=0

Always increasing

Both at y=0, y=1

No symettry

Square root function

f(x)=sqrtx

Starts from (0,0)

increasing always

Domain: [0, infinity)

Range: [0, infinity)