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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