CA TEST 1 - Chapters 1, 2, 5.1, 5.2

Percent Change

quantity changes from A to B; percent change is [(B-A)/A]x100.

The slope of a line has two formulas:

(y2-y1)/(x2-x1) and [f(x2)-f(x1)]/(x2-x1).

Scientific Notation

a number in the form (cx10^n), where 1 ≤ |c| ≤ 10.

Characteristics of a nonlinear function:

1. Graph is not a straight line. 2. Does not have a constant rate of change. 3. Cannot be written as f(x)=mx+b. 4. Can have any number of zeros. 5. Still can only have one value of y for every value of x.

Relation

A relation is a set of ordered pairs related in some way.

Interval Notation

A convenient notation allowing us to express a set of real numbers without having to draw the entire number line. Example: (x1, x2].

Symbolic Representation of F

A formula/equation gives a symbolic representation of a function.

Linear Function

A function f represented by f(x)=mx+b, where m and b are constants.

Function

A function is a relation in which no two x-values can have the same y-value; x-values cannot repeat.

Graphical Representation of F

A graph that visually pairs an x-input with a y-output in the form of ordered pairs plotted on a xy-plane.

Constant Function

A special type of linear function where f(x)=b, where b is a constant (fixed number). The slope is 0 because m=0.

Numerical Representation of F

A table of values that lists input-output pairs for a function. This is usually a partial numerical representation instead of a complete numerical representation.

One-to-one Function

Different inputs always produce/result in different outputs. That is, for elements c and d in the domain of f, c≠d implies f(c)≠f(d).

Function v. Relation

Every function is a relation, although not every relation is a function.

Repeated Element in a Domain or Range

If an element is repeated more than once in a domain or range, it is only listed once in the set for the domain or range.

Horizontal Line Test

If every horizontal line intersects the graph of a function f at most once, then f is a one-to-one function.

Vertical Line Test

If every vertical line intersects a graph at no more than one point, then the graph is a function.

f(x) = square root of (1-x)

If there is no sign (+/-) in front of the square root symbol, then the value of y (square root of (1-x)) is assumed to be positive.

IR

Means all real numbers.

Difference Quotient of f

Represents the average rate of change of f from x to x+h, meaning slope is [f(x+h)-f(x)]/h, where h≠0.

f^-1(x)

Represents the inverse of the function f and NOT a real number f raised to the -1th power. For instance, if f(x)=5x, then f^-1(x)=x/5.

Average Rate of Change

The average rate of change of a nonlinear function from x1 to x2 equals the slope of the line passing through points on its graph (x1, y1) and (x2, y2); slope=(y2-y1)/(x2-x1).

Standard Equation of a Circle

The circle with center (h, k) and radius r has equation (x-h)^2 + (y-k)^2 = r^2.

Composition of Functions

The composite of functions g and f is written gºf and is defined by (gºf)(x)=g)f(x)). We read this as "g of f of x."

Verbal Representation of F

The description of a function using words; " divide x seconds by 5 to obtain y miles."

Distance Formula

The distance between points (x1, y1), (x2, y2) in the xy-plane is d=the square root of [(x2-x1)^2 + (y2-y1)^2].

Domain of operations of f(x) and g(x)

The domain of any operation on the two functions f(x) and g(x) would give all of the x-values that are both in the domain of f(x) and g(x).

Set-Builder Notaion

The expressions {x|x≠1} and {y|1<y<5} are written in set-builder notation and represent 1) the set of all real numbers x such that x≠1 and 2) the set of all real numbers y such that y is greater than 1 and less than 5.

Open Interval

The inequality -1<x<4 does not include the endpoints -1, 4, therefore parentheses are used in the interval notation to signify this set is open: (-1, 4).

Closed Interval

The inequality -1≤x≤4 includes the endpoints -1, 4, therefore brackets are used in the interval notation to signify this set is closed: [-1, 4].

Secant Line

The line between the two points used to calculate the average rate of change of a nonlinear function.

Midpoint Formula

The midpoint of the line segment connecting (x1, y1), (x2, y2) is M = [(x1 + x2)/2, (y1 + y2)/2].

Domain

The set of x-values (first element) of a relation.

Range

The set of y-values (second element) of a relation.

Slope of a Line

The slope m of the line passing through the points (x1, y1) and (x2, y2) is m=(y2-y1)/(x2-x1), where x1≠x2. In other words, m=rise/run.

Median

The value that is located in the middle of a sorted list. If the list has odd amount of numbers, the median is the middle term. If there is an even amount, it is the average of the two middle items.

Finding a Symbolic Representation For f^-1

To find a formula for f^-1, perform the following steps: 1. Verify that f is a one-to-one function. 2. Solve the equation y=f(x) for x, obtaining the equation x=f^-1(y). 3. Interchange x and y to obtain y=f^-1(x).

Mean (Average)

To get the mean of a set of n numbers, you add the n numbers and then divide the sum by n.

Union Symbol

Used to write the inequality x<1 or x>3 in interval notation; it means "or" and looks like a U.

Applying Inverse Actions

We must reverse the order as well as apply the inverse operation at each step of inversing an action.

Multiplying Fractions

You multiply the two numerators together to get the new numerator, and then you multiply the two denominators together to get the new denominator.

Interval notation is not

an ordered pair. It is the domain of the x-values for a relation; the -1 and 4 in (-1, 4) are both values of x.

Inputs and outputs (domains and ranges)

are interchanged for inverse functions.

Equation of circle with center of (0, 0)

can be simplified from (x-0)^2 + (y-0)^2 = r^2 to x^2 + y^2 = r^2.

Points on the graph of f

can be written in the form of (x, f(x)) because f(x) = y.

Composition of functions is not

commutative.

Quadrants of a xy-plane are numbered

counterclockwise starting with the upper-right corner; quadrant I, II, III, IV.

An inverse function of f does not exist if

different inputs of a function f produce the same output.

Expression f(x+h) in difference quotient

does NOT equal f(x)+f(h).

Function Notation

f(x) = y is read "f of x equals y," meaning that function f with input x produces output y.

Every one-to-one function

has an inverse function.

Not every function

has an inverse function.

A function increases

if, whenever x1<x2, f(x1)<f(x2).

A function decreases

if, whenever x1<x2, f(x1)>f(x2).

The graph of f^-1

is a reflection of the graph of f across the line y=x.

The domain of gºf

is all x in the domain of f such that f(x) is in the domain of g.

A nonlinear function's rate of change

is always averaged; the average rate of change from points on the graph (x1, y1), (x2, y2).

A linear function's rate of change

is always constant (y/x).

The sum of two functions f and g

is defined by (f+g)(x)=f(x)+g(x).

The difference of two functions f and g

is defined by (f-g)(x)=f(x)-g(x).

The quotient of two functions f and g

is defined by (f/g)(x)=f(x)/g(x), where g(x)≠0.

The product of two functions f and g

is defined by (fg)(x)=f(x)•g(x).

A composition of f with itself

is denoted (fºf)(x).

Reading increasing/decreasing functions

is read from left to right.

The value of m in y=mx+b

is the slope of the line created when y=mx+b is graphed on the xy-plane. m=change in y/change in x, or rise/run.

The rate of change of a line

is the slope of the line.

The value of b in y=mx+b

is the y-coordinate of the y-intercept; (0, b).

A zero of the function f

is when any number c of any function f results in f(c)=0. An x-coordinate (c) of an x-intercept corresponds to an input that results in an output of 0.

For function increasing/decreasing statements

it is important to give x-intervals and these intervals DO NOT include the endpoints.

A point does not belong to a quadrant if

it lies on the x-axis or y-axis.

There are two types of functions:

linear and nonlinear

Exactly one zero is given per

linear function, provided the slope of the graph is not equal to 0 (m≠0).

(fºg)(x)

means f(g(x)) which means that the output (y) of g(x) is used as the input for f(x).

Dividing a fraction equals

multiplying the first fraction by the second fraction's reciprocal.

The composition of f^-1ºf and fºf^-1 with input x

produces output x.

To verify f^-1(x) is the inverse of f(x)

show that (f^-1ºf)(x)=x and (fºf^-1)(x)=x.

To get the reciprocal of a fraction

swap the numerator and the denominator; -3/4's reciprocal is -4/3. You keep the positive/negative aspect of the fraction throughout.

In mathematics, an inverse is associated with

the concept of reversing a calculation and arriving at the original value.

The range of f equals

the domain of f^-1.

If equal outputs of a function result in the same input

the function is one-to-one.

The domain of f equals

the range of f^-1.

If a continuous function f is always increasing on its domain

then every horizontal line will intersect the graph f at most once and f is a one-to-one function.

If a continuous function g is always decreasing on its domain

then every horizontal line will intersect the graph g at most once and g is a one-to-one function.

If a function is increasing on an interval

then larger input (x-values) produce larger outputs (y-values).

Inverse actions can

undo or cancel each other.

Number of y-values for every x-value of a function

will always be EXACTLY one y-value for every x-value. If this rule fails to be followed, the equation is no longer a function.

Independent Variable

x is the independent variable of a function.

Dependent Variable

y is the dependent variable of a function.

When multiplying a number by its reciprocal

you get 1.