Inverse Functions – Example

Minggu, 12 Januari 2020

Example 1 Given f (x) = 3x − 2 find f ⁻¹ (x).

Solution

Now, we already know what the inverse to this function is as we’ve already done some work with it. However, it would be nice to actually start with this since we know what we should get. This will work as a nice verification of the process.

So, let’s get started. We’ll first replace f (x) with y.

y = 3x − 2

Next, replace all x’s with y and all y’s with x.

x = 3y − 2

Now, solve for y.

       x + 2 = 3y

Finally replace y with f ⁻¹ (x).

Now, we need to verify the results. We already took care of this in the previous section, however, we really should follow the process so we’ll do that here. It doesn’t matter which of the two that we check we just need to check one of them. This time we’ll check that ( f ∘ f ⁻¹)(x) = x is true.

Example 2 Given find g⁻¹ (x).

Solution

Now the fact that we’re now using g (x) instead of  f (x) doesn’t change how the process works. Here are the first few steps.

Now, to solve for y we will need to first square both sides and then proceed as normal.

           x² = y – 3

                x² + 3 = y 

This inverse is then,

g⁻¹ (x) = x² + 3

Finally let’s verify and this time we’ll use the other one just so we can say that we’ve gotten both down somewhere in an example.

                                               (g ^{– 1} ∘  g) (x) = g ^{– 1}  [ g (x) ]                     

So, we did the work correctly and we do indeed have the inverse.

Example 3 Givenfind h⁻¹ (x) .

Solution

The first couple of steps are pretty much the same as the previous examples so here they are,

Now, be careful with the solution step. With this kind of problem it is very easy to make a mistake here.

x (2y – 5) = y + 4

 2xy – 5x = y + 4

     2xy – y = 4 + 5x

              (2x – 1) y = 4 + 5x             

             

So, if we’ve done all of our work correctly the inverse should be,

Finally we’ll need to do the verification. This is also a fairly messy process and it doesn’t really matter which one we work with.

Okay, this is a mess. Let’s simplify things up a little bit by multiplying the numerator and denominator by 2x − 1.

Wow. That was a lot of work, but it all worked out in the end. We did all of our work correctly and we do in fact have the inverse.


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Labels: Mathematician

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