**Chapter 1: Functions & Their Graphs**

Sections 5,6 & 8

Sections 5,6 & 8

**1.5 A Library of Functions**

By: Artur Tomaszkowicz

**1.8 Inverse Functions**

By: Caleb Rodriguez

**1.8 Inverse Functions cont...**

How to find an Inverse Function

**By: Artur Tomaszkowicz, Anayatzinc Vargas & Caleb Rodriguez**

This section consists of finding the inverse of functions and graphing them to see how they're similar and how they're different.

Constant Functions

**1.6 Shifting, Reflecting & Sketching Graphs**

Examples of Inverse Functions

**By: Anayatzinc Vargas**

**1.6 Shifting, Reflecting & Sketching Graphs**

**By: Anayatzinc Vargas**

f(x)=2x+1

f (x)=(x-1)/2

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This section will help you use vertical shifts, horizontal shifts, reflections, and nonrigid transformations to sketch graphs of functions.

-CITATIONS-

Chapter 1- Section 5

Chapter 1-Section 6

Chapter 1- Section 8

Shifting Graphs

Many functions have the simple transformations of the common graphs show section 1.5.

Example 1

h(x)=x +4 f(x)=x

This shifts the graph of f(p)=p^3 up four units.

Shifts are evident when you graph the original and new function together.

Vertical & Horizontal Shifts

Identity Function

The variable

a

is positive number. In a graph vertical and horizontal shifts of y=f(x) are shown below.

g(x)=√x

g (x)=x

Cubic Function

h(x)=3x-6

h (x)=1/3x+2

The vertical shift of a units upward :

h(x)=f(x)

+a

The vertical shift a units downward :

A constant function is a function with a domain of all real numbers and a range of a single number,

x

.

Horizontal line with the form

f(x)=c

f(x)=c

With the given equation, first we subtract 1 from each

side to get

x-1=2y

. Then, we divide both sides by 2. This

gives us the result,

y=(x-1)/2

.

Then, we plug it into the original

equations to make sure it is

correct. f(f (x))=2((x-1)/2)+1.

In the end, we get x=x.

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To solve this equation, first we square both sides to get

y=x . Then, when we plug the first equation into the

second equation, we get f(f (x))=√(x) . When we

solve this equation, we get

x=x.

2

h(x)=f(x)

-a

The horizontal shift a units to the right :

h(x)=f

(x-a)

Linear Functions

The horizontal shift a units to the left :

h(x)=f

(x+a)

Commonly shown as

f(x)=ax + b

, with

x

being the slope and

b

being the y-intercept.

Generally follow these rules.

Domain of function is set of all real numbers.

Range of function is set of all real numbers.

Has one intercept (0,

b

)

If increasing then

m

>0

If decreasing then

m

<0

If constant then

m

=0 ◘

A line with a domain and range that are a set of all real number. Has a slope of 1 and a

y

-intercept of (0,0).

Shown as

f(x)=x

.

f(x)=x

Combinations of these shifts are commonly found in graphs.

Vertical & Horizontal shifts generate a

family of functions

, having the same shape but in a different location.

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2

Squaring Functions

A parabola with a function of f(x)=x .

f(x)=x

Squaring Functions

Commonly shown as

f(x)=x .

Domain of function is set of all real numbers.

Range of function is set of all nonnegative real numbers.

Function is even.

Has one intercept (0,0)

Decreasing on interval ( -∞,0), and increasing on interval (0, ∞)

Symmetric with respect to y-axis.

Relative minimum at (0,0)

2

Vertical & Horizontal Shifts

2

It does not matter if the vertical shift comes before the horizontal shift or if the horizontal shift comes before the vertical shift. The same result is obtained.

Vertical & Horizontal Shifts

Example 2

This is an example of a combination of vertical & horizontal shifts.

Original

New

Example 4

The following is an example of a vertical shift

Original

New

Example 3

The following is an example of a horizontal shift

Original

New

Reflecting Graphs

The second most common type of transformation is a

reflection

.

New graphed function is a mirror image of the original graphed function.

The transformations of reflections and shifts can be combined as well in functions.

Reflecting Graphs

Reflections in the coordinate axis of the graph

y=f(x)

are represented below

1. Reflection in the x-axis

2. Reflection in the y-axis

h(x) = -f (x)

h(x) = f ( -x)

Reflecting Graphs

Functions can also have combinations of vertical and horizontal shifts with reflections.

Example 5

When you sketch the graphs involving square roots you need to remember that you must exclude negative numbers inside the radical.

Original

Reflection across axis

Nonrigid Transformations

This section will help you use vertical shifts, horizontal shifts, reflections, and nonrigid transformations to sketch graphs of functions.

Rigid

Nonrigid

Horizontal shifts

Vertical shifts

reflections

Basic shape of graph is unchanged

Only the position changes

Vertical stretch

Vertical shrink

Horizontal stretch

Horizontal shrink

Cause a distortion- a change in shape of the original graph

Nonrigid Transformations

Vertical Stretch

Vertical Shrink

Horizontal Stretch

Horizontal Shrink

c>1

0<c<1

c>1

0<c<1

g(x)= cf(x)

h(x)= f(cx)

y= f(x) is represented as the following shows

Nonrigid Transformations

Examples of each shift and shrink

vertical stretch

vertical shrink

horizontal stretch

horizontal shrink

2

**1.5 A Library of Functions**

A cubic function is generally shown as f(x)=x .

It follows these rules.

Domain of function is set of all real numbers.

Range of function is set of all real numbers.

Function is odd.

Has intercept at (0,0)

Increasing on interval (-∞,∞)

Graph is symmetric with respect to origin.

3

Square Root Function

A square root function is generally shown as f(x)=√x .

It follows these rules.

Domain of function is set of all nonnegative real numbers.

Range of function is set of all nonnegative real numbers.

Has intercept at (0,0)

Increasing on interval (0,∞)

Reciprocal Function

A cubic function is generally shown as f(x)= .

It follows these rules.

Domain of function is (-∞,0) or (0,∞).

Range of function is (-∞,0) or (0,∞).

Function is odd.

Does not have any intercepts

Decreasing on interval (-∞,0) and (0,∞)

Graph is symmetric with respect to origin.

1

x

_

only used information from textbook

came up with my own examples with http://rechneronline.de/function-graphs/

Step Function

A cubic function is generally shown as f(x)= .

It follows these rules.

Domain of function is set of all real numbers.

Range of function is set of all integers.

Function is odd.

Has y-intercept of (0,0) and x-intercept in the [0,0)

Graph is constant between each pair of consecutive integers.

Graph jumps vertically one unit at each integer value.

1

x

_

To find the inverse of this equation, first we add 6 to

both sides to get x+6=3y. Then, we divide both sides by

3 to get f (x)=x/3+2. When we plug in the inverse

equation into the original

equation, we get

f(f (x))=3(1/3x+2)-6.

After we solve, we get x=x.

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Textbook Information

Pictures from google

Created own examples

Pictures: www.google.com

Step 1: Use the horizontal Line Test to decide whether the equation has an inverse function.

Piecewise Function

A Piecewise function is two separate functions plotted on one coordinate plane.

This function has no inverse

This function has an inverse

Step 2: Replace the x and y values, then solve for the new y.

Ex: In the equation y=3x+1, we would change it into x=3y+1 and solve for y. The result would be

f (x)=x/3+1/3

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Step 3: Verify that f and f are inverse functions by plugging in one equation into the other. By showing that f(f (x))=x=f (f(x)), we can conclude that both equations are indeed inverses

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Absolute Value Function

An absolute value function is a function that contains an algebraic expression within absolute value symbols. Recall that the absolute value of a number is its distance from 0 on the number line. To graph an absolute value function, choose several values of x and find some ordered pairs.

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3

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