U2S6V03 What is an inverse function?.txt

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We gave you these two functions.
f was the function that cubes its input,
and g was the function that takes
the cube root of its input.
Now what do we mean when we say the cube root of y?
It's supposed to be the number x whose cube is y.
So cube root of y-- that should be a number, which
when we cube it, we get y back.
And similarly, if we have x cubed,
and we take the cube root of that, we'll get x.
Another way of saying this is if y equals x cubed, then
x is the cube root of y.
And conversely, if x is the cube root of y
then y will be equal to x cubed.
For instance, in the example, we told you
that minus 13 cubed equals negative 2,197.
And that's precisely the reason that the cube
root of negative 2,197 is negative 13.
If we use our function notation, f and g,
this here is saying that if y equals f of, x then x
equals g of y and vice versa.
And over here, the cube of the cube root of y is f of g of y.
And that equals y.
In other words, f undoes what g did.
And similarly, the cube root of the cube
of x, that can be written as g of f of x.
And that equals x.
g undoes what f did.
Functions that behave like this are called inverses.
We say that g is the inverse of f.
And we generally write it as g equals f minus 1 like this.
It looks like an exponent, but it's not.
So don't take f and raise it to the minus 1 power.
This just means that g is f inverse.
And similarly, f is going to be g's inverse.
And what f inverse does as a function is exactly
the same as the cube root.
f inverse of y is the number x such that f of x equals y.
So the input for f inverse is y, but that's
the desired output for f.
And the result of f inverse is x.
And that's the input for f.
So erasing all this stuff involving g and cube roots,
we have that y equals f of x exactly when x
is f inverse of y.
And then over here, f and f inverse
are going to undo one another.

One very important warning-- not every function has an inverse.
We'll discuss that a little bit later in the sequence.
But if it does have an inverse, this
is what the inverse function is supposed to do.
We have some questions to help you play around with it.