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# TAPPS Exercise: How Do I Solve That For y?

Module by: Kenny M. Felder. E-mail the author

Summary: This module provides a practice problem designed to lead students through a typical TAPPS problem: solving for y in an inverse function.

OK, so you’re looking for the inverse function of y = x 2x + 1 y = x 2x + 1 size 12{y= { {x} over {2x+1} } } {} and you come up with x = y 2y + 1 x = y 2y + 1 size 12{x= { {y} over {2y+1} } } {} . Now you have to solve that for yy size 12{y} {}, and you’re stuck.

First of all, let’s review what that means! To “solve it for y” means that we have to get it in the form yy size 12{y} {}=something, where the something has no yy size 12{y} {} in it anywhere. So y=2x+4y=2x+4 size 12{y=2x+4} {} is solved for yy size 12{y} {}, but y=2x+3yy=2x+3y size 12{y=2x+3y} {} is not. Why? Because in the first case, if I give you xx size 12{x} {}, you can immediately find yy size 12{y} {}. But in the second case, you cannot.

“Solving it for yy size 12{y} {}” is also sometimes called “isolating yy size 12{y} {}” because you are getting yy size 12{y} {} all alone.

So that’s our goal. How do we accomplish it?

 1. The biggest problem we have is the fraction. To get rid of it, we multiply both sides by 2y+12y+1 size 12{2y+1} {}. x ( 2y + 1 ) = y x ( 2y + 1 ) = y size 12{x $$2y+1$$ =y} {} 2. Now, we distribute through. 2 xy + x = y 2 xy + x = y size 12{2 ital "xy"+x=y} {} 3. Remember that our goal is to isolate yy size 12{y} {}. So now we get all the things with yy size 12{y} {} on one side, and all the things without yy size 12{y} {} on the other side. x = y − 2 xy x = y − 2 xy size 12{x=y - 2 ital "xy"} {} 4. Now comes the key step: we factor out a yy size 12{y} {} from all the terms on the right side. This is the distributive property (like we did in step 2) done in reverse, and you should check it by distributing through. x = y ( 1 − 2x ) x = y ( 1 − 2x ) size 12{x=y $$1 - 2x$$ } {} 5. Finally, we divide both sides by what is left in the parentheses! x 1 − 2x = y x 1 − 2x = y size 12{ { {x} over {1 - 2x} } =y} {}

Ta-da! We’re done! x12xx12x size 12{ { {x} over {1 - 2x} } } {} is the inverse function of x 2x + 1 x 2x + 1 size 12{ { {x} over {2x+1} } } {} . Not convinced? Try two tests.

Test 1:

Test 2:

Now, you try it! Follow the above steps one at a time. You should switch roles at this point: the previous student should do the work, explaining each step to the previous teacher. Your job: find the inverse function of x + 1 x 1 x + 1 x 1 size 12{ { {x+1} over {x - 1} } } {} .

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