Wondershare TidyMyMusic 2.1.0 add to watchlist send us an update. Buy now $39.00 $ 37.05 5% OFF! 4 screenshots: runs on: Windows 8 32/64 bit Windows 7 32/64 bit Windows Vista 32/64 bit. IMPLIES 1C2xC3x2C4x3CD 1.1 x/2 IMPLIES h1;2;3;4;:::i! 1.1 x/2: (12.1) We found a generating function for the sequence h1;2;3;4;:::iof positive inte-gers! In general, differentiating a generating function has two effects on the corre-sponding sequence: each term is multiplied by its index and the entire sequence is shifted left one place.

1. Tidymymusic 3 0 1 2 X 2.5
2. Tidymymusic 3 0 1 2 X 2x
3. Tidymymusic 3 0 1 2 X 2.25
4. Tidymymusic 3 0 1 2 X 28
5. Tidymymusic 3 0 1 2 X 2

An inverse function goes the other way!

Let us start with an example:

Here we have the function f(x) = 2x+3, written as a flow diagram: Haiku animation.

The Inverse Function goes the other way:

So the inverse of: 2x+3 is: (y-3)/2

The inverse is usually shown by putting a little '-1' after the function name, like this:

f^-1(y)

We say 'f inverse of y'

So, the inverse of f(x) = 2x+3 is written:

f^-1(y) = (y-3)/2

(I also used y instead of x to show that we are using a different value.)

Back to Where We Started

The cool thing about the inverse is that it should give us back the original value:

When the function f turns the apple into a banana,
Then the inverse function f^-1 turns the banana back to the apple

Example:

Using the formulas from above, we can start with x=4:

f(4) = 2×4+3 = 11

We can then use the inverse on the 11:

f^-1(11) = (11-3)/2 = 4

And we magically get 4 back again!

We can write that in one line:

f^-1( f(4) ) = 4

'f inverse of f of 4 equals 4'

So applying a function f and then its inverse f^-1 gives us the original value back again:

f^-1( f(x) ) = x

We could also have put the functions in the other order and it still works:

f( f^-1(x) ) = x

Example:

Start with:

f^-1(11) = (11-3)/2 = 4

And then:

f(4) = 2×4+3 = 11 Quiver 3 0 3 download free.

So we can say:

f( f^-1(11) ) = 11

'f of f inverse of 11 equals 11'

Solve Using Algebra

We can work out the inverse using Algebra. Put 'y' for 'f(x)' and solve for x:

This method works well for more difficult inverses.

Fahrenheit to Celsius

A useful example is converting between Fahrenheit and Celsius:

For you: see if you can do the steps to create that inverse!

Inverses of Common Functions

It has been easy so far, because we know the inverse of Multiply is Divide, and the inverse of Add is Subtract, but what about other functions?

Install brave browser on mac. Here is a list to help you:

(Note: you can read more about Inverse Sine, Cosine and Tangent.)

Careful!

Did you see the 'Careful!' column above? That is because some inverses work only with certain values.

Example: Square and Square Root

When we square a negative number, and then do the inverse, this happens: Radium 3 0 5.

But we didn't get the original value back! We got 2 instead of −2. Our fault for not being careful!

So the square function (as it stands) does not have an inverse

But we can fix that!

Restrict the Domain (the values that can go into a function).

Example: (continued)

Just make sure we don't use negative numbers.

In other words, restrict it to x ≥ 0 and then we can have an inverse.

So we have this situation:

• x² does not have an inverse
• but {x² | x ≥ 0 } (which says 'x squared such that x is greater than or equal to zero' using set-builder notation) does have an inverse.

No Inverse?

Let us see graphically what is going on here:

To be able to have an inverse we need unique values.

Just think . Jump desktop 7 1 2. if there are two or more x-values for one y-value, how do we know which one to choose when going back?

Imagine we came from x₁ to a particular y value, where do we go back to? x₁ or x₂?

In that case we can't have an inverse.

But if we can have exactly one x for every y we can have an inverse.

It is called a 'one-to-one correspondence' or Bijective, like this

A function has to be 'Bijective' to have an inverse.

So a bijective function follows stricter rules than a general function, which allows us to have an inverse.

Domain and Range

So what is all this talk about 'Restricting the Domain'?

In its simplest form the domain is all the values that go into a function (and the range is all the values that come out).

As it stands the function above does not have an inverse, because some y-values will have more than one x-value.

But we could restrict the domain so there is a unique x for every y .

Note also:

• The function f(x) goes from the domain to the range,
• The inverse function f^-1(y) goes from the range back to the domain.

Let's plot them both in terms of x . so it is now f^-1(x), not f^-1(y):

f(x) and f^-1(x) are like mirror images
(flipped about the diagonal).

In other words:

The graph of f(x) and f^-1(x) are symmetric across the line y=x

Example: Square and Square Root (continued)

First, we restrict the Domain to x ≥ 0:

• {x² | x ≥ 0 }'x squared such that x is greater than or equal to zero'
• {√x | x ≥ 0 }'square root of x such that x is greater than or equal to zero'

And you can see they are 'mirror images'
of each other about the diagonal y=x.

Note: when we restrict the domain to x ≤ 0 (less than or equal to 0) the inverse is then f^-1(x) = −√x:

• {x² | x ≤ 0 }
• {−√x | x ≥ 0 }

Tidymymusic 3 0 1 2 X 2.5

Which are inverses, too.

Not Always Solvable!

It is sometimes not possible to find an Inverse of a Function.

Example: f(x) = x/2 + sin(x)

We cannot work out the inverse of this, because we cannot solve for 'x':

y = x/2 + sin(x)

y . ? = x

Notes on Notation

Even though we write f^-1(x), the '-1' is not an exponent (or power):

Summary

The inverse of f(x) is f^-1(y)
We can find an inverse by reversing the 'flow diagram'
Or we can find an inverse by using Algebra:
• Put 'y' for 'f(x)', and
• Solve for x
We may need to restrict the domain for the function to have an inverse

How to describe a set by saying what properties its members have.

A Set is a collection of things (usually numbers).

Example: {5, 7, 11} is a set.

But we can also 'build' a set by describing what is in it.

Google chrome per windows vista download gratis. Here is a simple example of set-builder notation:

It says 'the set of all x's, such that x is greater than 0'.

In other words any value greater than 0

Notes:

• The 'x' is just a place-holder, it could be anything, such as { q | q > 0 }
• Some people use ':' instead of '|', so they write { x : x > 0 }

Type of Number

It is also normal to show what type of numberx is, like this:

• The means 'a member of' (or simply 'in')
• The is the special symbol for Real Numbers.

So it says:

'the set of all x's that are a member of the Real Numbers,
such that x is greater than or equal to 3'

In other words 'all Real Numbers from 3 upwards'

There are other ways we could have shown that:

On the Number Line it looks like:

In Interval notation it looks like: [3, +∞)

Number Types

We saw (the special symbol for Real Numbers). Here are the common number types:

Example: { k | k > 5 }

'the set of all k's that are a member of the Integers, such that k is greater than 5'

In other words all integers greater than 5.

This could also be written {6, 7, 8, . } , so:

{ k | k > 5 } = {6, 7, 8, . }

Why Use It?

When we have a simple set like the integers from 2 to 6 we can write:

{2, 3, 4, 5, 6}

But how do we list the Real Numbers in the same interval?

{2, 2.1, 2.01, 2.001, 2.0001, . ???

So instead we say how to build the list:

{ x | x ≥ 2 and x ≤ 6 }
Start with all Real Numbers, then limit them between 2 and 6 inclusive.

We can also use set builder notation to do other things, like this:

{ x | x = x² } = {0, 1}
All Real Numbers such that x = x²
0 and 1 are the only cases where x = x²

Another Example:

Example: x ≤ 2 or x > 3

Set-Builder Notation looks like this:

{ x | x ≤ 2 or x >3 }

On the Number Line it looks like:

Using Interval notation it looks like:

(−∞, 2] U (3, +∞)

We used a 'U' to mean Union (the joining together of two sets).

Defining a Domain

Set Builder Notation is very useful for defining domains.

In its simplest form the domain is the set of all the values that go into a function.

The function must work for all values we give it, so it is up to us to make sure we get the domain correct!

Example: The domain of 1/x

1/x is undefined at x=0 (because 1/0 is dividing by zero).

So we must exclude x=0 from the Domain:

The Domain of 1/x is all the Real Numbers, except 0

We can write this as

Dom(1/x) = {x | x ≠ 0}

Example: The domain of g(x)=1/(x−1)

1/(x−1) is undefined at x=1, so we must exclude x=1 from the Domain:

The Domain of 1/(x−1) is all the Real Numbers, except 1

Using set-builder notation it is written:

Dom( g(x) ) = { x | x ≠ 1}

Example: The domain of √x

Tidymymusic 3 0 1 2 X 2x

Is all the Real Numbers from 0 onwards, because we can't take the square root of a negative number (unless we use Imaginary Numbers, which we aren't).

We can write this as

Dom(√x) = {x | x ≥ 0}

Example The domain of f(x) = x/(x² − 1)

To avoid dividing by zero we need: x² - 1 ≠ 0

Factor: x² - 1 = (x−1)(x+1)

(x−1)(x+1) = 0 when x = 1 or x = −1, which we want to avoid!

Tidymymusic 3 0 1 2 X 2.25

So:

Tidymymusic 3 0 1 2 X 28

Tidymymusic 3 0 1 2 X 2

Dom( f(x) ) = {x | x ≠ 1, x ≠ −1}