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Finding inverses of rational functions

Video transcript

alright let's say that we have the function f of X and it's equal to 2x plus 5 over 4 minus 3x and what we want to 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 and this is a reminder of what a function and 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 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 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 2x plus 5 over 4 minus 3x 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 2y plus 5 over 4 minus 3y and then to be able to express this as a function of x to say that what is Y is 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 4 minus 3y if we do that on the left hand side we are going to get x times each of these terms so we're you get 4x minus 3y X and then that's going to be equal to on the right-hand side since we multiplied by the denominator here we're going to be left with the numerator it's going to be equal to 2y plus 5 and this could be a little bit intimidating because we're seeing it X's and Y's 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 2y here actually so I could go either way let's get rid of this 2y here so let's subtract 2y from both sides and let's get rid of this 4x from the left-hand side so let's subtract 4x from both sides and then what are we going to be left with on the left-hand side we're left with minus or negative 5 or actually it would be this way it would be negative 3y X minus 2y 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 going to have 5 minus 4x 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 3x minus 2 is equal to 5 minus 4x and now this is the homestretch we can just divide both sides of this equation by negative 3x minus 2 and we're going to get Y is equal to 5 minus 4 x over negative 3x minus 2 now another way that you could express this you could multiply both the numerator in the denominator by a negative 1 that won't change the value and then you would get you would get in the numerator of 4x minus 5 and in the denominator you would get a 3x plus 2 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