About this video
The function y(x) equals the cube root of x cubed plus one. Composing it with itself, y(y(x)), means feeding the whole output back in as the input. Cubing undoes the cube root, leaving x cubed plus one, and then the outer function adds one more, so y(y(x)) equals the cube root of x cubed plus two.
You'll be able to
- evaluate a function at an expression rather than a number
- explain why cubing a cube root gives back the original quantity
- compose a function with itself and simplify the result
Key words
- Function
- A rule that gives exactly one output for each input.
- Composition
- Feeding the output of one function into another function, written f(g(x)).
- Cube root
- The number that, when cubed, gives the original value.
- Evaluate
- To find the output of a function for a given input.
- Input and output
- The value put into a function and the value that comes out.
Transcript
- [Teacher] Let y of x be equal to the cube root of x to the third plus one. Which of the following is equivalent to y of y of x? All right, this is gonna be fun. So we know that y of x is equal to the cube root of x to the third plus one. Which of the following is equivalent to y of y of x?
Well, to evaluate y of y of x, y of, y of y of x, everywhere where we see an x here, we will just replace it with a y of x. And it's kind of, it might seem a little bit daunting 'cause it's referring to itself but it should all make sense in the end. So, we're gonna have something to the third power plus one, and in this case, that something is y of x.
y of x. So, what's y of x equal to? Well, it's equal to all of these business. So, let me write this down. So this is going to be equal to the cube root of, let me give myself some space. So, y of x is the cube root of x to the third plus one. Now we're going to raise that to the third power.
We'll raise that to the third power plus one. And so we just have to evaluate this. So, what is this going to be? Well, this is going to be the cube root of, well, what we have right over here. If I take the cube root of something but then I raise it to the third power, this is like raising something to the 1/3 and then raising that to the third.
Well, that's just gonna give me this thing. This, all of this business right over here, is just going to evaluate to x to the third plus one. Once again, if I just take the cube root of something, of whatever this is, and then raise that to the third power, that's going to be this original something.
Maybe I should put a, if I take the cube root of a and if I raise it to the third power, that's just going to be a here. That's all I did right over here. The cube root of x to the third plus one to the third power is just gonna be x to the third plus one. And then now, we have the plus one, so this is going to be equal to the cube root of x to the third plus two.
x to the third plus two, which is that choice right over there.
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