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  • - [Instructor] All right, let's say that we have

  • the function f of x and it's equal to two x plus five,

  • over four minus three x.

  • And what we wanna do is figure out what

  • is the inverse of our function.

  • Pause this video and try to figure

  • that out before we work on that together.

  • All right, now let's work on it together.

  • Just as a reminder of what a function

  • and an inverse even does,

  • if this is the domain of a function

  • and that's the set of all values that you could input

  • into the function for x and get a valid output.

  • And so let's say you have an x here,

  • it's a member of the domain.

  • And if I were to apply the function to it,

  • or if I were to input that x into that function.

  • Then the function is going to output a value

  • in the range of the function and we call that value f of x.

  • Now an inverse, that goes the other way.

  • If you were to input the f of x value into the function

  • that's going to take us back to x.

  • So that's exactly what f inverse does.

  • Now how do we actually figure out the inverse

  • of a function especially a function

  • that's defined with a rational expression like this.

  • Well the way that I think about it is,

  • let's say that y is equal to our function of x

  • or y is a function of x so we could say

  • that y is equal to two x plus five,

  • over four minus three x.

  • For our inverse the relationship

  • between x and y is going to be swapped.

  • And so in our inverse it's going

  • to be true that x is going to be equal to two y plus five,

  • over four minus three y.

  • And then to be able to express this

  • as a function of x, to say that what is y as a function

  • of x for our inverse we now have to solve for y.

  • So it's just a little bit of algebra here.

  • So let's see if we can do that.

  • So the first thing that I would do

  • is multiply both sides of this equation by four minus 3 y.

  • If we do that, on the left hand side we are going

  • to get x times each of these terms.

  • So we're going to get four x minus three yx

  • and then that's going to be equal to

  • on the right hand side, since we multiplied

  • by the denominator here we're just going

  • to be left with the numerator.

  • It's going to be equal to two y plus five.

  • And this could be a little bit intimidating

  • 'cause we're seeing xs and ys, what are we trying to do,

  • remember we're trying to solve for y.

  • So let's gather all the y terms on one side

  • and all the non-y terms on the other side.

  • So let's get rid of this two y here.

  • Actually, well I could go either way.

  • Let's get rid of this two y here,

  • so let's subtract two y from both sides.

  • And let's get rid of this four x from the left hand side,

  • so let's subtract four x from both sides.

  • And then what're we going to be left with.

  • On the left hand side we're left with minus

  • or negative five, or actually it would be this way,

  • it would be negative three yx minus two y.

  • And you might say hey where is this going,

  • but I'll show you in a second,

  • is equal to, those cancel out

  • and we're gonna have five minus four x.

  • Now once again we are trying to solve for y.

  • So let's factor out a y here,

  • and then we are going to have y,

  • times negative three x minus two

  • is equal to five minus four x.

  • And now this is the homestretch.

  • We can just divide both sides of this equation

  • by negative three x minus two

  • and we're going to get y is equal

  • to five minus four x, over negative three x minus two.

  • Now another way that you could express this

  • is you could multiply both the numerator

  • and the denominator by negative one,

  • that won't change the value.

  • And then you would get, you would get

  • in the numerator four x minus five,

  • and in the denominator you would get a three x plus two.

  • So there you have it.

  • Our f inverse as a function of x,

  • which we could say is equal to this y

  • is equal to this right over there.

- [Instructor] All right, let's say that we have

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尋找有理函數的反函數|方程|代數2|Khan Academy (Finding inverses of rational functions | Equations | Algebra 2 | Khan Academy)

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    林宜悉 發佈於 2021 年 01 月 14 日
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