1. 1 Mod 2
2. Capto 1 2 16 Equals Equal
3. 1=2 Proof

An inverse function goes the other way!

1/2 times what equals 1/2? What divided by 1/4 equals 3/5? What is 2 over 3 as a percentage? How to calculate 3/4 plus 1/9 Is 1/2 greater than 1/3? What is 4.56 as a fraction? How to calculate 3/7 divided by 4/5 Equivalent Fractions of 3/9 What is the factorial of 6? 1/2 divided by what equals 3/4? Reduce 5/25 What times 1/3 equals 1/2? To share our screencasts with viewers, we need to export a finished video file out of Capto. To export a video, go to File Export. Most video-sharing services prefer mp4 video files. Capto's export options are limited, but should be enough for most purposes. For simplicity, I usually let Capto create my mp4 videos. 1=2: A Proof using Beginning Algebra The Fallacious Proof: Step 1: Let a=b. Step 2: Then, Step 3:, Step 4:, Step 5:, Step 6:. Step 7: This can be written as, Step 8: and cancelling the from both sides gives 1=2. See if you can figure out in which step the fallacy lies.

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

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:

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:

(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:

• 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?

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 }

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

1 Cup

'1 Cup' is equal to 8 fluid ounces in US Standard Volume. It is a measure used in cooking.

A Metric Cup is slightly different: it is 250 milliliters (which is about 8.5 fluid ounces).

(In italics: not exact)

Measuring Cups

Measuring cups are used to measure liquids, or powders like flour or sugar.

As we fill up the measuring cup, we need to look at which number it is filled to.

Here we have 2/3 cup (two-thirds of a cup) of milk.

And they can also measure milliliters (ml):

1 Mod 2

Here we have 150 ml.

It doesn't say '150' ... it says '50' ... but it is half-way between 100 and 200 so we can figure out it is 150 ml.

There are also special ones designed to hold an exact amount, such as these 1/2 cup and 1/4 cup measures.

Capto 1 2 16 Equals Equal

They can also hold more than '1 Cup' of liquid.

This measuring cup holds up to 4 cups of liquid. If the liquid goes all the way to the 3, then there are 3 cups of liquid. Between the numbers we find fractions of a cup.

1=2 Proof

2 cups of a liquid is the same thing as 1 pint. So this measuring cup also holds 2 pints (4 cups equals 2 pints).