Mathlify

Composing inverse functions

We can also compose a function with its inverse to get composite functions ff−1 and f−1f.

We can try to find the formula for f−1(x) and perform composition, but turns out we can get the result in a simpler manner by consideration the meaning of this composition.

f−1 is the function that reverses f, so f−1f will take a number, apply f to it and then try to reverse the process. We then end up with the original number. A similar argument works for ff−1, leading us to the following results.

Formulas

  • ff−1(x)=x
  • f−1f(x)=x

Differences between the two functions

Is ff−1(x) and f−1f(x) the exact same function? Turns out that is not true because of domain considerations. Recall that Dfg=Dg so we have

  • Df−1f=Df
  • Dff−1=Df−1

Using the result

Question:

f:x↦2x−1,x∈ℝ,g:x↦x2+3,x∈ℝ.

We also know that f−1(x)=x+12.
Solve

fg(x)=5.

Give a definition, in similar form, for the composite function fg.

Discussion:

We could tackle the question by finding a formula for the composite function fg(x).

However, if we already have a formula for f−1(x), we can get the answer in a simpler way by applying the result f−1f(x)=x.

In particular, we use the result f−1fg(x)=g(x) in the following solution.

Solution:

fg(x)=7Applying f−1 on both sides,f−1fg(x)=f−1(5)Note that f−1fg(x)=g(x),g(x)=f−1(5)x2+3=5+12x2+3=3x=0∎