Example 1. Let
(Assume the function is one-to-one function.) Find f-1(x). First
we set and solve for x.
We see and raise the exponents
to 4 both sides yields,
Finally, we write for (The reason we restrict is to make f-1 a
one-to one function, otherwise f-1 will not have an inverse.) Let's
verify this inverse by following two methods.
Method 1. (Checking f(g(x))=g(f(x))=x).
You could use the following Maple commands to check your answer:
f:=x -> 2*x^(1/4);
g:=x -> (x/2)^4;
simplify(f(g(x)));
simplify(g(f(x)));
Method 2. Check if the graphs are symmetric to y=x.
f:= x -> 2*x^(1/4);
g:=x -> (x/2)^4;
plot({f(x),g(x),x},x=0..10,y=0..10, scaling=constrained);
You should get something like this .
Example 2. Let for This function is one-to-one
function if why?). Find f-1(x).
First we set and solve for x.
Take the square root both sides, yieds, Finaly, we write Can you use two methods mentioned above to
verify indeed f-1 is the inverse of f?