Example:Let f:R→R
be defined by f(x)=3x+1
. Find the inverse function f −1
.
Solution: For any input x
, the function machine corresponding to f
spits out the value y=f(x)=3x+1
. We want to find the function f −1
that takes the value y
as an input and spits out x
as the output. In other words, y=f(x)
gives y
as a function of x
, and we want to find x=f −1 (y)
that will give us x
as a function of y
.
To calculate x
as a function of y
, we just take the expression y=3x+1
for y
as a function of x
and solve for x
.
yy−1y−13 =3x+1=3x=x
Therefore, we found out that x=y/3−1/3
, so we can write the inverse function as
f −1 (y)=y3 −13 .
In the definition of the function f −1
, there's nothing special about using the variable y
. We could use any other variable, and write the answer as f −1 (x)=x/3−1/3
or f −1 (★)=★/3−1/3
. The placeholder variable used in the formula for a function doesn't matter.
To verify that f −1
is really the inverse of f
, we should show that the composition of f
and f −1
doesn't do anything to the input. In this case, the order shouldn't matter, and the functions f∘f −1
and f −1 ∘
f$ should both do nothing. Let's check this.
First, we apply f
followed by f −1
.
(f −1 ∘f)(x) =f −1 (f(x))=f −1 (3x+1)=(3x+1)/3−1/3=x+1/3−1/3=x
Second, we apply f −1
followed by f
.
(f∘f −1 )(x) =f(f −1 (x))=f(x/3−1/3)=3(x/3−1/3)+1=x−1+1=x
In both cases, applying both f
and f −1
to x
gave us back x
. Indeed, f −1 (x)=x/3−1/3
.
No comments:
Post a Comment