1st Nine Weeks AP Pre-Calculus Exam Review

Function

a relation (set of ordered pairs)/mapping for which each value/element of the domain is mapped/paired to exactly one value/element of the range

Relation

a general term for a set of ordered pairs

What is true of a relation?

any set of ordered pairs where elements in the domain may/may not be paired with all or some of the elements in the range; can be any subset of the domain and the range of a function, may not necessarily be a function

Piecewise Function

consists of a set of functions defined over non-overlapping domain intervals

What can piecewise functions be used for?

modeling a data set or contextual scenario that demonstrates different characteristics over different intervals

Domain

the set of all possible values of the independent variable

Range

the set of all possible values of the dependent variable

Linear Parent Function

f(x)=x

Domain restriction for linear functions

all real numbers

Range restriction for linear functions

all real numbers

Domain restriction for cubic functions

all real numbers

Range restriction for cubic functions

all real numbers

Linear and cubic functions are odd, so what type of functions have no domain or range restrictions?

odd functions

Quadratic function domain restriction

all real numbers

Quadratic function range restriction

[0, ∞)

How do you find the domain of a square root function?

set the radicand greater than or equal to zero and solve for x

What happens if what is under the radicand is anything other than a linear expression?

you must do a sign/sketch test

How do you find the range of a square root function?

use your knowledge of transformations to see if the function has been reflected across the ex-axis and/or been shifted up or down vertically

What is the range if a>0?

[k, ∞)

What is the range if a<0?

(-∞,k]

Rational functions

when there is a function of x in the denominator

Rational function domain restrictions

the denominator cannot equal zero

How do you find the domain of a rational function?

set the denominator equal to zero and solve for x; then when you get x, this is what x should NOT equal

Range of a quadratic function with positive coefficient

[min/max, ∞)

Range of a quadratic function with a negative coefficient

(-∞, min/max]

What equation gives the value for which a function can be maximized or minimized?

h=-b/2a

What equation gives the max or min value?

k=c-b²/4a

Domain restriction of exponential functions

all real numbers

Range restriction for exponential functions if a>0

(k,∞)

Range restriction for exponential functions if a<0

(-∞,k)

What kind of asymptotes do exponential functions have?

horizontal asymptotes

How do you find horizontal asymptotes on exponential functions?

y=k

How do you find the domain of a logarithmic function?

set the argument (what is in the parentheses) greater (>) than zero and solve for x; if what is in the argument is any expression other than a linear one, you must do a sign/sketch test

Range restriction of a logarithmic function

all real numbers

Greatest Integer Function

f(x)=[[x]]

Domain restriction of greatest integer function

all real numbers

Range of greatest integer function

{y:yεZ}

Domain restriction of absolute value function

all real numbers

Range restriction of absolute value function if a>0

[k,∞)

Range restriction of absolute value function if a<0

(-∞,k]

Set Notation

the set of all x or y values, such that, and then state the domain or range

What do greater/less than or equal to signs mean?

makes the values listen inclusive (included in the domain or range)

Interval Notation

use parentheses and/or brackets to state the domain or range

Brackets

value is included in the domain

Parentheses

value is NOT included in the domain

When is a function linear?

if the average rate of change over any length input value is CONSTANT

Linear functions must have…

constant rate of change, first difference is constant, and the second difference is zero

When is a function quadratic?

if the average rate of change over consecutive equal-length input value intervals is LINEAR

Quadratic functions must have…

a linear rate of change, first difference is linear, change in the average rate of change is constant, second difference is constant, third difference is zero

Formula for Average Rate of Change

f(b)-f(a)/b-a

Slope of a secant line

m=y2-y1/x2-x1

Points of inflection

where f(x) changes concavity; when the rates of change of f(x) change from increasing to decreasing or decreasing or increasing

f(x) is increasing when…

the input values increase, the output values increase; if a<b, f(a)<f(b)

f(x) is decreasing when…

the input values increase when the output values decrease; a>b, then f(a)>f(b)

When is a function positive?

if the output values are always positive, meaning the graph of f(x) is above the x-axis; f(x)>0

When is a function negative?

if the output values are always negative, meaning the graph of f(x) is below the x-axis; f(x)<0

If the rate of f(x) is increasing…

f(x) is concave up

If the rate of f(x) is decreasing…

f(x) is concave down

If the rate of f(x) is positive…

f(x) is increasing

If the rate of f(x) is negative…

f(x) is decreasing

Even Functions

f(-x)=f(x)

How do you graph even functions?

reflect across the y-axis

Odd Functions

f(-x)=-f(x)

How do you graph odd functions?

reflect across the origin (rotational symmetry about the origin of 180 degrees) or reflect over the x-axis AND the y-axis

When is a function invertible?

if each output value of f(x) is mapped from a unique input value

What is h when optimizing quadratics?

the value for which the quadratic function is maximized/minimized

What is k when optimizing quadratics?

the max/min value

If even degreed polynomial…

same end behaviors

If odd degreed polynomial…

opposite end behaviors