Question

PLEASE!!! NEED HELP ASAP!!! 50 PTS AND BRAINLIEST!!!

1) There are at least 3 ways to show that 2 functions are inverses. However, only ONE method is a valid proof. Although Methods A and B might convince you that the functions are inverses, they are NOT acceptable methods to verify that functions are inverses. Complete the work for each method below:

Method A: Given () = 3 − 4 and () = +4/3
Show that these are inverse functions by finding ^−1 () and showing that it is the same as ().

Method B: Given () = 3 − 4 and () = +4/3
Show that these are inverse functions by showing that when the output of one function is used for the input of the other function, the final output is equal to the original input value. (you may choose any initial input)

Method C: Given () = 3 − 4 and () = +4/3
Verify that these are inverse functions by showing that (()) = AND (()) =

Answer 1

The notation that we use really depends upon the problem. In most cases either is acceptable.

For the two functions that we started off this section with we could write either of the following two sets of notation.

f(x)=3x−2f−1(x)=x3+23g(x)=x3+23g−1(x)=3x−2f(x)=3x−2f−1(x)=x3+23g(x)=x3+23g−1(x)=3x−2

Now, be careful with the notation for inverses. The “-1” is NOT an exponent despite the fact that is sure does look like one! When dealing with inverse functions we’ve got to remember that

f−1(x)≠1f(x)f−1(x)≠1f(x)

This is one of the more common mistakes that students make when first studying inverse functions.

The process for finding the inverse of a function is a fairly simple one although there is a couple of steps that can on occasion be somewhat messy. Here is the process

That’s the process. Most of the steps are not all that bad but as mentioned in the process there are a couple of steps that we really need to be careful with.

In the verification step we technically really do need to check that both (f∘f−1)(x)=x(f∘f−1)(x)=x and (f−1∘f)(x)=x(f−1∘f)(x)=x are true. For all the functions that we are going to be looking at in this section if one is true then the other will also be true. However, there are functions (they are far beyond the scope of this course however) for which it is possible for only of these to be true. This is brought up because in all the problems here we will be just checking one of them. We just need to always remember that technically we should check both.

There is one final topic that we need to address quickly before we leave this section. There is an interesting relationship between the graph of a function and its inverse.

Here is the graph of the function and inverse from the first two examples. We’ll not deal with the final example since that is a function that we haven’t really talked about graphing yet.

In both cases we can see that the graph of the inverse is a reflection of the actual function about the line y=xy=x. This will always be the case with the graphs of a function and its inverse.

https://tutorial.math.lamar.edu/classes/alg/inversefunctions.aspx