Inverse functions (§5.2)
As composition involves performing one function after another,
so inverses involve performing a function backwards.
Two functions f and g
are inverse functions of each other if:
Not every function has an inverse function!
If f does have an inverse function, then it only has one;
we call f one-to-one
and denote its unique inverse function by f−1.
- f(g(x)) = x
whenever x ∈ dom g, and
- g(f(x)) = x
whenever x ∈ dom f.
f−1 does not mean 1/f!
There are two ways to caclulate f−1:
If you ever get more than one solution when you solve the equation,
then f is not one-to-one, and it has no inverse.
- Start with x = f(y) and solve for y;
this gives you y = f−1(x).
- Start with y = f(x) and solve for x;
this gives you x = f−1(y).
- f−1(f(x)) = x
whenever x ∈ dom f, and
- f(f−1(y)) = y
whenever y ∈ ran f.
The graphs of inverse functions are related by switching x and y.
A function is one-to-one
if and only if its graph satisfies the Horizontal Line Test:
every horizontal line goes through the graph at most once.
- The range of f
is the same as the domain of f−1;
- the range of f−1
is the same as the domain of f.
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