0.5 A discus thrower accelerates a discus from rest to a speed of 25.3 m/s by whirling it through 1.28 rev. Assume the discus moves on the arc of a circle 0.99 m in radius. a) Calculate the nal angular speed of the discus. b) Determine the magnitude of the angular acceleration of the discus, assuming it to be constant. Visit ESPN to view the 2020 NFL Standings. Ryan Clark breaks down how Le'Veon Bell helps the Chiefs incorporate a new aspect to their already stacked offense. M8 thread SACC-E-MU-M 8 1504071 100 SACC-E-MU-M 8 1504071 100 M8 flush-type socket M8 flush-type plug PHOENIX CONTACT 225 M5 to M12 circular plug-in connectors M5/M8 sensor/actuator device plug-in connectors M8, 3, 4, 5-pos. M8, front mounting, with 0.5 m long litz wires. What's new in HoudahSpot 6.0.4: HoudahSpot is now available on Setapp: a subscription service for Mac and iOS apps. For $9.99/mo, use HoudahSpot plus over 200 other apps. No extra fees, no ads. Fixes a crash when organizing search criteria snippets in HoudahSpot preferences; Fixes a bug with filtering. HoudahSpot Upgrades. HoudahSpot 5.0 is a paid upgrade from previous versions. The single user upgrade license is priced at $ 19.The family upgrade license is priced at $ 29.
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:
The Inverse Function goes the other way:
So the inverse of: 2x+3 is: (y-3)/2 Winzip 6 5 – mac edition of established compression utility.
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
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:
| The function: | f(x) | = | 2x+3 |
| Put 'y' for 'f(x)': | y | = | 2x+3 |
| Subtract 3 from both sides: | y-3 | = | 2x |
| Divide both sides by 2: | (y-3)/2 | = | x |
| Swap sides: | x | = | (y-3)/2 |
| Solution (put 'f-1(y)' for 'x') : | f-1(y) | = | (y-3)/2 |
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?
Here is a list to help you:
| Inverses | Careful! | ||
| <=> | |||
| <=> | Don't divide by zero | ||
| 1x | <=> | 1y | x and y not zero |
| x2 | <=> | x and y ≥ 0 | |
| xn | <=> | or | n not zero (different rules when n is odd, even, negative or positive) |
| ex | <=> | ln(y) | y > 0 |
| ax | <=> | loga(y) | y and a > 0 |
| sin(x) | <=> | sin-1(y) | -π/2 to +π/2 |
| cos(x) | <=> | cos-1(y) | 0 to π |
| tan(x) | <=> | tan-1(y) | -π/2 to +π/2 |
(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:
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:
Houdahspot 5
- 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 .. if there are two or more x-values for one y-value, how do we know which one to choose when going back?
| General Function |
| No Inverse |
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
Houdahspot 5 0 11 Msds Sheet
| Bijective Function |
| Has an Inverse |
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'?
Houdahspot 5 0 11 Mean
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 }
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):
| f-1(x) | ..is different to.. | f(x)-1 |
| Inverse of the function f | f(x)-1 = 1/f(x) (the Reciprocal) |
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
In Big Sur 11.1, Apple introduced an annoying system bug that made Open and Save dialogs revert to a very small size every time you used them. To see anything in the dialog, you pretty much had to resize them. Every. Single. Time. Default Folder X 5.5.4 brought a fix for this – forcing the dialogs back to your preferred, larger size whenever they came up.
Houdahspot 5 0 11 Miles
Fast forward to Big Sur 11.2, and Apple has fixed the bug so dialogs now stay larger once you resize them – yay! Well, mostly yay anyway. The sidebar still bounces back to its narrow, 'I can't read the names of my folders' width of about 100 pixels every second time you use an Open or Save dialog.
So here's Default Folder X 5.5.6. If you resize the sidebar in an Open or Save dialog, Default Folder X will make sure it bounces back to that size the next time you use the same file dialog.
In addition, this release expands support for HoudahSpot so that you now get the 'Search in HoudahSpot' menu item in Default Folder X's utility menu if you're using the version of HoudahSpot included in the Setapp subscription service. It previously only worked with copies of HoudahSpot bought directly from Houdah Software.
- {x2 | x ≤ 0 }
- {−√x | x ≥ 0 }
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):
| f-1(x) | ..is different to.. | f(x)-1 |
| Inverse of the function f | f(x)-1 = 1/f(x) (the Reciprocal) |
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
In Big Sur 11.1, Apple introduced an annoying system bug that made Open and Save dialogs revert to a very small size every time you used them. To see anything in the dialog, you pretty much had to resize them. Every. Single. Time. Default Folder X 5.5.4 brought a fix for this – forcing the dialogs back to your preferred, larger size whenever they came up.
Houdahspot 5 0 11 Miles
Fast forward to Big Sur 11.2, and Apple has fixed the bug so dialogs now stay larger once you resize them – yay! Well, mostly yay anyway. The sidebar still bounces back to its narrow, 'I can't read the names of my folders' width of about 100 pixels every second time you use an Open or Save dialog.
So here's Default Folder X 5.5.6. If you resize the sidebar in an Open or Save dialog, Default Folder X will make sure it bounces back to that size the next time you use the same file dialog.
In addition, this release expands support for HoudahSpot so that you now get the 'Search in HoudahSpot' menu item in Default Folder X's utility menu if you're using the version of HoudahSpot included in the Setapp subscription service. It previously only worked with copies of HoudahSpot bought directly from Houdah Software.
And because someone asked for it, holding down the Option key while selecting a Folder Set from Default Folder X's menu in your menu bar will open all of your Favorite folders in the Finder. So if you've got multiple Folder Sets for different projects or workflows, you can now instantly open all of those folders in the Finder when you're starting work. (A few of you old-timers might remember that this was actually a feature in version 4 that didn't make the jump to v5 – now it's back!)
There are also a few bug fixes in version 5.5.6, as I'm still chasing after the elusive problem of the cursor sporadically disappearing on some Macs (but not others). Full release notes and download links are on the Default Folder X Release Page. Or just choose 'Check for Updates' from Default Folder X's menu if you're already running it.
