Unit 1 - Functions: Characteristics and Properties

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Unit 1 - Functions: Characteristics and Properties

1.4 - Properties of Graphs of Functions (GC)

Turning Points

A point on a graphical curve where the function changes from an increase to decrease or vice versa; which is from decrease to an increase

Graphical Characteristics

A function can be categorized from its graphical characteristics. Given a set, the type of functions that has certain graphical characteristics can be determined and be eliminated from the rest

X and Y Intercepts

Continuity/Discontinuity

Intervals of Increae/Decreae

Turning Point

Symmetry (Even/Odd)

Domain And Range

Asymptotes

An asymptote is an "imaginary" line that gets closer and closer to but will never cross on some part of its domain

Domain for f(x) = 1/f(x): { xer / x cannot equal 0}

End Behaviours

Behaviour of a f(x) graph, when x either approach positive infinity or negative infinity.

The degree and leading coefficient determine the end behaviour of a graph.

Negative End Behaviour: where it is approaching negative infinity

Positive End Behaviour; where it is approaching positive infinity

Continuous & Discontinuous Functions

Discontinuous Function

a function that has holes, vertical asymptotes, or any jumps present in the function

The break in the graph is called an asymptote or a point of discontinuity

Continuous Function

a function that does not contain any breaks or holes over its entire domain/x values.

Symmetry

Neither Odd or Even

in order to make sure if the function given is neither odd nor even, you have to input both the even function and the off function.

Odd Functions

f(-x) = -f(x)

Graphing Odd Functions

"symmetric about the origin" - half of the graph on one side of the y-axis is upside down compared to the other half of the graph on the other side of the y-axis.

Even Functions

Algebraically:

f(-x) = f(x)

Graphing Even Functions

"symmetric about the y-axis" - whatever the graph is doing on one side of the y-axis is reflected on the other side of the y-axis

Intervals

1.3 - Sketching Graphs of Functions

Transformations

y = + or -af[k(x+ or - d)] + or - c

In y = f(-x), there is reflection in the y axis

In y = -f(x), there is a reflection in the x axis

In y = af(x), there is a vertical stretch or compression by a units

In y = f(kx), there is a horizontal stretch or compression by k units

In fy = (x) + or - c. a vertical translation by c units

In f(x+ or - d), there is a horizontal translation by d units

must be performed in a particular order: 01. horizontal/vertical stretches/compressions 02. reflections 03. horizontal/vertical translations (shifts

Subtopic

Restrictions

Rather than state all the numbers that are possible for the domain and range, the symbol R is used to classify all real are included, and afterwards state the numbers that are not possible afterwards. These are called restrictions

Range

set of all possible y-values of an equation given

set of all possible x-values of an equation given

1.2 - Exploring Absolute Values

Absolute Value & Inequalities

a > 0 ; a is positive

all the points that are less than a and are greater than - a is included in this number line

[2, ∞)

( -∞, -a]

if l x l ≥ a, then - a ≤ x ≥ a

if l x l > a, then -a < x > a

all the points that are less than a but greater than - a is included in this number line

Brackets

(-3, 3)

if l x l ≤ a, then -a ≤ x ≤ a

closed circles are used when it comes to this number line since the point is a part of the solution

l x l < a ; then - a < x < a

open circles are used when it comes to the number line since the point is not a part of the solution

Absolute Value

f(x) = l x l

If you want to seek the absolute value of a number, whether positive or negative, it is always going to be a positive number (with a few exceptions). When you evaluate, it is possible to be involved with operations.

It is a distance, not a direction. This is the reason why it cannot be negative.

1.1 - Relations & Functions

How to Solve:

Vertical Line Test on a Graph

If it passes through it twice, then the graph is not a functoin

If it passes the vertical line test, than one line drawn vertically through the graph should only be touching that one line.

Through Ordered sets:

Not a Function

Function

Functions

relation/set of ordered pairs in which for every balie for x, there is only one definite value for y. y cannot be shared by two x values, it only has to be shared by 1.

function notation

tool that is used to represent the value of (Y) for a given value of x .

x-y notation

Relations

a set of ordered pairs in which an x value is paired with a y value.

1.8 - Composite Functions

Two or more functions combined results in what is called a "Composite Function"

To solve: given both f(x) and g(x), find what the question is asking and then later simplify your answers.

f(g(x))

this notation is slightly more usable than the other one because it allows you to see that you will first work inside of the function

fog (x)

1.7 - Exploring Operations with Functions

Domain

If two functions have the same domain, they can be added or multiplied to get a new function on that shared domain.

Given coordinates:

If given a set of coordinates. You only look at the sets of coordinates in both f(x) and g(x) to see if both of the share any of the same x values. If they do, simply proceed to add/subtract/multiply only the ones that share the same domain.

To solve algebraically:

If given the equations, preferably two equations in the form of f(x) and in the form of g(x), you can do whatever operation (except division) that the question is stating to do.

You can add/subtract.multiply the expressions and then simplify

To solve graphically:

If given a graph, only the y values are being manipulated. The y values are only manipulated when the y values share the same domain. If the domain is the same, then you can add/subtract/multiply.

1.6 - Piecewise Functions

Piecewise functions are functions that are represented by two or more components, or "pieces". The separate component of a piecewise function is defined for a specific interval in the domain of a function

To Determine Algebraic Representation:

1. Determine the closed circles and open circles 2. Figure out what parent function the component is 3. Solve.

To Graph:

1. Draw each of the graphs given in your equation 2. Make small lines on the places you have to "cut" each of the three graphs. At this time, also figure out if the graphs given is closed circle or open circle. This can give you an idea on if the graph is going to be continuous or not. 3. Assemble all the graphs together!

When a piecewise function is graphed, it be a continuous graph or a discontinuous graph.

The Domain of piecewise functions can vary. - If it is continuous, it has a domain of all real number. - If discontinuous, there is a break in the graph and there are a few restrictions on the domain.

1.5 - Inverse Relations

Domain & Range

Domain of the original function will be the range of the inverse function. The range of the original function will be the domain of the inverse function.

Inverse Functions

Self-Inverse Function

When the inverse of the function is the same as the original function. The function is symmetrical about the line of y - x

Inverse Function

A function (that passes the vertical line test) that is a reflection of y - f(x) in the line of y - x

Finding The Inverse

Graphically

Interchange the values for x and y. (a, b) on the f(x) graph will be (b, a) on the graph of f^-1(x).

Algebraically

Given the equation for f(x), replace the x with y and isolate for y.