Function inverses
Function Inverses Example 2 Function Inverses Example 2
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- We've got the function f of x is equal to x plus 2 squared
- plus 1, and we've constrained our domain that x has to be
- greater than or equal to negative 2.
- That's where we've defined our function.
- And we want to find its inverse.
- And I'll leave you to think about why we had to constrain
- it to x being a greater than or equal to negative 2.
- Wouldn't it have been possible to find the inverse if we had
- just left it as the full parabola?
- I'll leave you -- or maybe I'll make a future video about that.
- But let's just figure out the inverse here.
- So, like we've said in the first video, in the
- introduction to inverses, we're trying to find a mapping.
- Or, if we were to say that y -- if we were to say that y is
- equal to x plus 2 squared plus 1.
- This is the function you give me an x and it maps to y.
- We want to go the other way.
- We want to take, I'll give you a y and then map it to an x.
- So what we do is, we essentially just solve
- for x in terms of y.
- So let's do that one step at a time.
- So, the first thing to do, we could subtract 1 from both
- sides of this equation.
- y minus 1 is equal to x plus 2 squared.
- And now to solve here, you might want to
- take the square root.
- And that actually will be the correct thing to do.
- But it's very important to think about whether you want
- to take the positive or the negative square
- root at this step.
- So we've constrained our domain to x is greater than or
- equal to negative 2.
- So this value right here, x plus 2, if x is always greater
- than or equal to negative 2, x plus 2 will always be
- greater than or equal to 0.
- So this expression right here, this right here is positive.
- This is positive.
- So we have a positive squared.
- So if we really want to get to the x plus 2 in the appropriate
- domain, we want to take the positive square root.
- And in the next video or the video after that, we'll solve
- an example where you want to take the negative square root.
- So we're going to take theundefined positive square
- root, or just the principal root, which is just the square
- root sign, of both sides.
- So you get the square root of y minus 1 is equal to x plus 2.
- And one thing I should have remembered to do is, from
- the beginning we had a constraint on x.
- We had for x is greater than or equal to negative 2.
- But what constraint could we have on y?
- If you look at the graph right here, x is greater than
- equal to negative 2.
- But what's why?
- What is the range of y-values that we can get here?
- Well, if you just look at the graph, y will always be
- greater than or equal to 1.
- And that just comes from the fact that this term right
- here is always going to be greater than or equal to 0.
- So the minimum value that the function could take on is 1.
- So we could say for x is greater than or equal to
- negative 2, and we could add that y is always going to be
- greater than or equal to 1.
- y is always greater than or equal to 1.
- The function is always greater than or equal
- to that right there.
- To 1.
- And the reason why I want to write it at the stage is
- because, you know, later on, we're going to swap
- the the x's and y's.
- So let's just leave that there.
- So here we haven't explicitly solved for x and y.
- But we can write for y is greater than or equal to 1,
- this is going to be the domain for our inverse, so to speak.
- And so here we can keep it for y is greater
- than or equal to 1.
- This y constraint's going to matter more.
- Because over here, the domain is x.
- But for the inverse, the domain is going to be the y-value.
- And then, let's see.
- We have the square root of y minus 1 is equal to x plus 2.
- Now we can subtract 2 from both sides.
- We get the square root of y minus 1 minus 2, is equal
- to x for y is greater than or equal to 1.
- And so we've solved for x in terms of y.
- Or, we could say, let me just write it the other way.
- We could say, x is equal to, I'm just swapping this. x is
- equal to the square root of y minus one minus 2, for y is
- greater than or equal to one.
- So you see, now, the way we've written it out.
- y is the input into the function, which is going to be
- the inverse of that function.
- x the output. x is now the range.
- So we could even rewrite this as f inverse of y.
- That's what x is, is equal to the square root of y minus 1
- minus 2, for y is greater than or equal to 1.
- And this is the inverse function.
- We could say this is our answer.
- But many times, people want the answer in terms of x.
- And we know we could put anything in here.
- If we put an a here, we take f inverse of a.
- It'll become the square root of a minus 1 minus 2, 4.
- Well, assuming a is greater than or equal to 1.
- But we could put an x in here.
- So we can just rename the the y for x.
- So we could just do a renaming here.
- So we can just rename y for x.
- And then we would get -- let me scroll down a little bit.
- We would f inverse of x.
- I'll highlight it here.
- Just to show you, we're renaming y with x.
- You could rename it with anything really, is equal to
- the square root of x minus 1.
- Of x minus 1.
- Minus 2 for, we have to rename this to, for x being
- greater than or equal to 1.
- And so we now have our inverse function as a function of x.
- And if we were to graph it, let's try our best to graph it.
- Maybe the easiest thing to do is to draw some points here.
- So the smallest value x can take on is 1.
- If you put a 1 here, you get a 0 here.
- So the point 1, negative 2, is on our inverse graph.
- So 1, negative 2 is right there.
- And then if we go to 2, let's see, 2 minus 1 is 1.
- The principle root of that is 1.
- Minus 2.
- So it's negative 1, so the point, 2, negative
- 1 is right there.
- And let's think about it.
- Let's see.
- If we did 5, I'm trying take perfect squares.
- 5 minus 1 is 4, minus 2.
- So the point 5, 2 is, let me make sure.
- 5 minus 1 is 4.
- Square root is 2.
- Minus 2 is 0.
- So the point 5, 0 is here.
- And so the inverse graph, it's only defined for x greater
- than or equal to negative 1.
- So the inverse graph is going to look something like this.
- It's going to look something like, I started off
- well, and it got messy.
- So it's going to look something like that.
- Just like that.
- And just like we saw, in the first, the introduction to
- function inverses, these are mirror images around
- the line y equals x.
- Let me graph y equals x. y equals x.
- y equals x is that line right there.
- Notice, they're mirror images around that line.
- Over here, we map the value 0 to 5.
- If x is 0, it gets mapped to 5.
- Here we go the other way.
- We're mapping 5 to -- we're mapping 5 to the value 0.
- So that's why they're mirror images.
- We've essentially swapped the x and y.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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