Finding the inverse of a function
unity, identity, inverses, composition of functions, square root
Inverses are defined as a part of a set which when multiplied by another part of a set give us the identity. There are many examples of these.For instance:
2*(1/2)=1 Since 1 is the Identity or Unity in multiplication, it means that since 2 brings (1/2) to unity, 2 is the inverse of (1/2) and vice versa.
An inverse is denoted by a symbol that looks like a negative 1 power. Along these lines, for polynomial functions, the unity or Identity I(x)=x Thus, inverses in this field are those that when multiplied by together give us this unity.
Inverses of polynomials are denoted by f-1(x).
Since the idea of multiplication is captured in the same way as the real numbers by composition of functions in polynomial functions, that is precisely where we find our profound definition of what it means to be an inverse.
if f(x) and f-1(x) are inverses of one another, we would get that
f(f-1(x))=I(x)
or f-1(f(x))=I(x)
This means that f( f-1(x))=I(x)=x
so, to find the inverse of x2, we do the following:
Since f(x)=x2 here, we would like to get:
f(f-1(x))=x
(f-1(x))2=x
To solve this, we take the square root of both sides, learning that:
(f-1(x))=
x
Thus, we see that the inverse of f(x)=x2 is (f-1(x))=
x.
All other computations of inverses of polynomial functions follow the same outline
unity, identity, inverses, composition of functions, square root