Chapters 2&3

Basic functions

Absolute Value Function

f(x)=|x|

Square Root Function

f(x)=√x

Cube Root function

f(x)=3√x

Identity Function

f(x)=x

Squaring Function

f(x)=x^2

Cubing Function

f(x)=x^3

Transformations

f (x) + b shifts the function b units upward

f (x) − b shifts the function b units downward

f (x + b) shifts the function b units to the left

f (x − b) shifts the function b units to the right

−f (x) reflects the function in the x-axis

f (−x) reflects the function in the y-axis

Arithmetic Operations on Functions

Addition

(f+g)(x)

Subtraction

(f−g)(x)

Multiplication

(fg)(x)

Division

(f/g)(x)

Composition of Functions and Domain

Standard Function:
(f∘g)(x)=f(g(x))

Domain

Find the domain of g

Find the domain of f

You must find the inputs x within the domain of g for which g(x) is within the domain of f

Specifically, exclude any input x from the domain of g for which g(x) is not in the domain of f

The resulting set is the domain of f∘g

Polynomial Functions

Find the zeros for the polynomial

Plot the x- and y-intercepts on the coordinate plane

Determine which way the ends of the graph point

Determine whether the graph lies above or below the x-axis by picking any value between consecutive x-intercepts and plugging it into the function

Plot the graph

rational Functions

Find the intercepts

Find the vertical asymptotes

Find the horizontal asymptote, if there is one

vertical asymptotes will divide the number line into regions

Plot graph

Linear functions

Distance equation:
d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2

Midpoint Equation:
M = ( x 1 + x 2 2 , y 1 + y 2 2 )

Standard Form:
Ax + By = C

Applications of Linear Equation

Geometry Problems

Motion Problems

Mixture Problems

Application of linear equations