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Composition of Functions:
     Inverse Functions and Composition (page 6 of 6)

Sections: Composing functions that are sets of point, Composing functions at points, Composing functions with other functions, Word problems using composition, Inverse functions and composition

The lesson on inverse functions explains how to use function composition to verify that two functions are inverses of each other. However, there is another connection between composition and inversion:

  • Given f (x) = 2x – 1 and
    g    (x) = (1/2)x + 4    ,
    find     f –1(x), g –1(x), ( f o g)–1(x),
    and      (g–1 o f –1)(x).

    What can you conclude?

    • This involves a lot of steps, so I'll stop talking and just show you how it goes.

    First, I need to find f –1(x), g –1(x), and ( f o g)–1(x):

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    Inverting  f (x):

      f (x) = 2x – 1
          y = 2x – 1
          y + 1 = 2x
          (y + 1)/2 = x
          (x + 1)/2 = y
          (x + 1)/2 =  f –1(x)

    Inverting g(x):

      g(x) = (1/2)x + 4
          y = (1/2)x + 4
          y – 4 = (1/2)x
          2(y – 4) = x
          2y – 8 = x
          2x – 8 = y
          2x – 8 = g –1(x)

    Finding the composed function:   Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

      ( f o g)(x) = f (g(x)) = f ((1/2)x + 4)
          = 2((1/2)x + 4) – 1
          = x + 8 – 1
          = x + 7

    Inverting the composed function:

      ( f o g)(x) = x + 7
          y = x + 7
          y – 7 = x
          x – 7 = y
          x – 7 = ( f o g)–1(x)

    Now I'll compose the inverses of  f(x) and g(x) to find the formula for (g–1 o f –1)(x):

      (g–1 o f –1)(x) = g–1( f –1(x))
          = g–1( (x + 1)/2 )
          = 2( (x + 1)/2 ) – 8
          = (x + 1) – 8
          = x – 7 = (g–1 o f –1)(x)

    Note that the inverse of the composition (( f o g)–1(x)) gives the same result as does the composition of the inverses ((g–1 o f –1)(x)). So I would conclude that

      ( f o g)–1(x) = (g–1 o f –1)(x)

While it is beyond the scope of this lesson to prove the above equality, I can tell you that this equality is indeed always true, assuming that the inverses and compositions exist — that is, assuming there aren't any problems with the domains and ranges and such.

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Cite this article as:

Stapel, Elizabeth. "Inverse Functions and Composition." Purplemath. Available from
    https://www.purplemath.com/modules/fcncomp6.htm. Accessed
 

 

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