Function Composition, Function Inverses, and How They Relate
Purplemath
One use of function composition is for checking if two functions are inverses of each other. If you compose the two functions and end up with just x, then the functions are inverses of each other. The lesson on inverse functions explains and demonstrates how this works.
However, there is another connection between function composition and function inversion. You may be expected to show (or to "discover") this connection by doing some symbolic computations.
- Given f(x) = 2x − 1 and , find the following:
- f −1(x)
- g−1(x)
- (f ∘ g)−1(x)
- (g−1 ∘ 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 ∘ g)(x). Then I can find (f ∘ g)−1(x) and g−1(x) ∘ f −1(x).
Inverting f (x):
f (x) = 2x − 1
y = 2x − 1
y + 1 = 2x
Inverting g(x):
Finding the composed function:
Inverting the composed function:
(f ∘ g)(x) = x + 7
y = x + 7
y − 7 = x
x − 7 = y
x − 7 = (f ∘ g)−1(x)
Now I'll compose the inverses of f (x) and g(x) to find the formula for (g−1 ∘ f −1)(x):
Note that the inverse of the composition (( f ∘ g)−1(x)) gives the same result as does the composition of the inverses ((g−1 ∘ f −1)(x)). So I would conclude that:
(f ∘ g)−1(x) = (g−1 ∘ f −1)(x)
Having completed this exercise, I notice that the order of f and g is reversed. What if I'd composed the inverses in the other order?
This is clearly not equal to the inverse of the composition f ∘ g, so clearly the order of the inverted functions in the composition *must* be opposite the order of the composed functions in the inverse.
What is the relationship between the inverse of a composition and the composition of the inverses?
When you have two invertible functions, the inverse of the composition of these functions is equal to the composition of the inverses of the functions, but in the reverse order. In other words, given f (x), g(x), and their composition (f ∘ g)(x), all invertible, then:
(f ∘ g)−1(x) = g−1(x) ∘ f −1(x)
I'll say it again: The order of the functions is reversed in the composition of the inverses from the order they'd been in the inverse of the composition. Don't forget this detail!
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, you can invert the composition by composing the inverses, but in the reverse order.
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