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Properties of Function Notes

This document discusses functions including determining if a relation is a function, types of functions, transforming functions using parameters, and the inverse of functions. It covers topics such as polynomial, absolute value, square root, sinusoidal, and inverse variation functions as well as vertical and horizontal scale changes in transformations.

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117 views11 pages

Properties of Function Notes

This document discusses functions including determining if a relation is a function, types of functions, transforming functions using parameters, and the inverse of functions. It covers topics such as polynomial, absolute value, square root, sinusoidal, and inverse variation functions as well as vertical and horizontal scale changes in transformations.

Uploaded by

welcometotheswamp
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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General Functions

1 - Determining if a Relation is a Function

A relation is NOT a function if:

a vertical line can be drawn anywhere


that intersects the graph more than once
2 – Types (Families) of Functions

Polynomial Functions
0-degree Polynomial Function 1st-degree Polynomial Function 2nd-degree Polynomial Function

Functions Resulting from Operations


Absolute-Value Function Square-Root Function

Inverse Variation Function Exponential Function

Special Functions
Periodic Function Step Function Piecewise Function

Piecewise Functions
−4 𝑥 < 8
𝑓(𝑥 ) = & −√−2𝑥 8 ≤ 𝑥 < 0
−3𝑥 𝑥 ≥ 0

3 Transforming Functions Using PARAMETERS

PARAMETERS are numbers that modify the shape of


the graph of a function.

The “multiplicative parameters” are:

“a” is the “vertical scale change”.

“b” is the “horizontal scale change”.


a) Representation of Transformations using
Function Rules

A TRANSFORMED rule looks like this:

g(x) = a • f (b • x )

Function Basic Rule Transformed


Rule
1st Degree f(x) = x f(x) = a(bx)
2nd Degree f(x) = x2
Exponential f(x) = cx
Step (G.I.F.) f(x) = [ x ]
Absolute-Value f(x) = I x I
Square-Root f(x) = Öx
Sinusoidal f(x) = sin (x)
Inverse Variation f(x) = 1
x
Function Basic Rule Transformed
Rule
1st Degree f(x) = x f(x) = a(bx)
2nd Degree f(x) = x2 f(x) = a(bx)2
Exponential f(x) = cx f(x) = acbx
Step (G.I.F.) f(x) = [ x ] f(x) = a[ bx ]
Absolute-Value f(x) = I x I f(x) = a I bx I
Square-Root f(x) = Öx f(x) = aÖbx
Sinusoidal f(x) = sin (x) f(x) = a sin (bx)
Inverse Variation f(x) = 1 f(x) = a
x bx

b) Coordinate Representation of Transformations

Rule for mapping points from a basic function to


corresponding image points of a transformed
function:
𝒙
( 𝒙, 𝒚 ) → ( , 𝒂𝒚+
𝒃
Example:

The ordered pair ( 8 , 8 ) associated with function f(x)


corresponds to ( 2 , 4 ) associated with function g(x).
Function g(x) is obtained by transforming f(x) using
only multiplicative parameters.

What are the values of these parameters?


𝒙
( 𝒙, 𝒚 ) → ( , 𝒂𝒚+
𝒃
𝟖
( 𝟖, 𝟖 ) → - , 𝒂 × 𝟖/ = (𝟐, 𝟒)
𝒃

𝟖
= 𝟐 & 𝒂 × 𝟖 = 𝟒
𝒃

𝒃 = 𝟒 & 𝒂 = 𝟎. 𝟓
c) Graphical Representation of Transformations

Note: in the following examples:

the original graph is of f(x) (shown in blue)

the transformed graph is of g(x) (shown in red)

Vertical Scale Change (a)

1<a<∞

0<a<1
-1 < a < 0
a = -1
-∞ < a <-1
Inverse of Functions

The graphs of function f and its inverse f-1 are


symmetrical about the 1st diagonal
(of the Cartesian plane)
Inverse of Functions

The graphs of function f and its inverse f-1 are


symmetrical about the 1st diagonal
(of the Cartesian plane)

Transformation
f( x ) ® f-1 ( x )
=(x,y)® (y,x)

The inverse of a
parabola is NOT a
function.

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