Function Inverse Example 1 Function Inverse Example 1
Function Inverse Example 1
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- So we have f of x is equal to negative x plus 4, and f of x
- is graphed right here on our coordinate plane.
- Let's try to figure out what the inverse of f is.
- And to figure out the inverse, what I like to do is I set y, I
- set the variable y, equal to f of x, or we could write that y
- is equal to negative x plus 4.
- Right now, we've solved for y in terms of x.
- To solve for the inverse, we do the opposite.
- We solve for x in terms of y.
- So let's subtract 4 from both sides.
- You get y minus 4 is equal to negative x.
- And then to solve for x, we can multiply both sides of this
- equation times negative 1.
- And so you get negative y plus 4 is equal to x.
- Or just because we're always used to writing the dependent
- variable on the left-hand side, we could rewrite this as x is
- equal to negative y plus 4.
- Or another way to write it is we could say that f
- inverse of y is equal to negative y plus 4.
- So this is the inverse function right here, and we've written
- it as a function of y, but we can just rename the y as x
- so it's a function of x.
- So let's do that.
- So if we just rename this y as x, we get f inverse of x is
- equal to the negative x plus 4.
- These two functions are identical.
- Here, we just used y as the independent variable, or
- as the input variable.
- Here we just use x, but they are identical functions.
- Now, just out of interest, let's graph the inverse
- function and see how it might relate to this
- one right over here.
- So if you look at it, it actually looks
- fairly identical.
- It's a negative x plus 4.
- It's the exact same function.
- So let's see, if we have-- the y-intercept is 4, it's going
- to be the exact same thing.
- The function is its own inverse.
- So if we were to graph it, we would put it right
- on top of this.
- And so, there's a couple of ways to think about it.
- In the first inverse function video, I talked about how a
- function and their inverse-- they are the reflection
- over the line y equals x.
- So where's the line y equals x here?
- Well, line y equals x looks like this.
- And negative x plus 4 is actually perpendicular to y is
- equal to x, so when you reflect it, you're just kind of
- flipping it over, but it's going to be the same line.
- It is its own reflection.
- Now, let's make sure that that actually makes sense.
- When we're dealing with the standard function right
- there, if you input a 2, it gets mapped to a 2.
- If you input a 4, it gets mapped to 0.
- What happens if you go the other way?
- If you input a 2, well, 2 gets mapped to 2 either
- way, so that makes sense.
- For the regular function, 4 gets mapped to 0.
- For the inverse function, 0 gets mapped to 4.
- So it actually makes complete sense.
- Let's think about it another way.
- For the regular function-- let me write it explicitly down.
- This might be obvious to you, but just in case it's
- not, it might be helpful.
- Let's pick f of 5.
- f of 5 is equal to negative 1.
- Or we could say, the function f maps us from 5 to negative 1.
- Now, what does f inverse do?
- What's f inverse of negative 1?
- f inverse of negative 1 is 5.
- Or we could say that f maps us from negative 1 to 5.
- So once again, if you think about kind of the sets, they're
- our domains and our ranges.
- So let's say that this is the domain of f, this
- is the range of f.
- f will take us from to negative 1.
- That's what the function f does.
- And we see that f inverse takes us back from negative 1 to 5.
- f inverse takes us back from negative 1 to 5, just
- like it's supposed to do.
- Let's do one more of these.
- So here I have g of x is equal to negative 2x minus 1.
- So just like the last problem, I like to set y equal to this.
- So we say y is equal to g of x, which is equal to
- negative 2x minus 1.
- Now we just solve for x.
- y plus 1 is equal to negative 2x.
- Just added 1 to both sides.
- Now we can divide both sides of this equation by negative 2,
- and so you get negative y over 2 minus 1/2 is equal to x, or
- we could write x is equal to negative y over 2 minus 1/2, or
- we could write f inverse as a function of y is equal to
- negative y over 2 minus 1/2, or we can just rename y as x.
- And we could say that f inverse of-- oh, let me careful here.
- That shouldn't be an f.
- The original function was g , so let me be clear.
- That is g inverse of y is equal to negative y over 2 minus 1/2
- because we started with a g of x, not an f of x.
- Make sure we get our notation right.
- Or we could just rename the y and say g inverse of x is equal
- to negative x over 2 minus 1/2.
- Now, let's graph it.
- Its y-intercept is negative 1/2.
- It's right over there.
- And it has a slope of negative 1/2.
- Let's see, if we start at negative 1/2, if we move over
- to 1 in the positive direction, it will go down half.
- If we move over 1 again, it will go down half again.
- If we move back-- so it'll go like that.
- So the line, I'll try my best to draw it, will
- look something like that.
- It'll just keep going, so it'll look something like that, and
- it'll keep going in both directions.
- And now let's see if this really is a reflection over y
- equals x. y equals x looks like that, and you can see
- they are a reflection.
- If you reflect this guy, if you reflect this blue line, it
- becomes this orange line.
- But the general idea, you literally just-- a function
- is originally expressed, is solved for y in terms of x.
- You just do some algebra.
- Solve for x in terms of y, and that's essentially your inverse
- function as a function of y, but then you can rename
- it as a function of x.
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