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Finding an Inverse Algebraically - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

Once we learn the definition of a function's inverse we learn how to find the algebraic inverse, or how to find the inverse using algebraic methods. There are different methods for finding the inverse, the most common of which is to switch the dependent and independent variables and solve for the dependent variable. This is an important step in learning how to prove the inverse of a function.

Finding the inverse of a funtion Algebraically. So for this particular example, so what we want to do is find an equation for a inverse function. We're given a function here. In this case we know that our equation is a line. 3x-2 we know that's a line therefore we know it's 1 to 1 and it's going to have an inverse.
So how we find a inverse is by switching our x and our y, for this particular problem we don't have a y but what we need to remember is f of x is the same thing as y. So just by replacing f of x with y, 3x-2, I now have an equation using x and y. To find the inverse we switch x and y's, okay? So every time we see an x, we throw in a y, every time we see a y we throw in an x. Okay? And we're used to seeing equations of y equals. So really from here all we want to do is solve for y. Okay?
So do that as we would any other equqtion, add 2 to both sides x+2=3y divided by 3, x+2 divide by 3 is equal to y. And then we found y but we're asked to find f inverse of x. Once we switch our variables solve for y, this is now the inverse okay? So we can replace our y with our inverse notation. x+2 over 3. Okay?
So to find the inverse, it's quite simple. So make sure you if you have an f of x replace it with y. solve through sorry. Switch your x and y's. Solve through for y making sure to replace it with your inverse notation at the end.
Now some teachers do this a little different and I just want to touch on that for a second. You don't have to switch x and y right here, okay? What you could do is actually solve for this x and then switch it. In general, I tend to shy away from that just because I'm so used to solving for y where what I've done is I've done problems where I haven't switched it. Halfway down the problem I forget that I'm solving for x, go back solve for y and I end up with exactly what I started with. Okay?
But if you can keep track what you're solving for, that's perfectly acceptable. For me, I always switch it. Fairly one of my first steps solve for y and then just make sure you replace your inverse notation.

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