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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?

Table 1
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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