Matlab cant find the inverse of my function

My matlab cant find the inverse of this function

f =

It results in "warning: Unable to find functional inverse" and "ans = Empty syms: 0-by-1"

I've tried to remove the a variable of a and change it into a number, but the result is still the same

syms x a real
f = sqrt((a+1)/(a-1))*atan(sqrt((a-1)*(x^2-1)/(a+1)))-atan(sqrt(x^2-1))

f = 

finverse(f, a)

Warning: Unable to find functional inverse.

 
ans =
 
Empty sym: 0-by-1
 

[Dyuman Joshi]

Are you sure that an inverse exists?

[Walter Roberson]

Not every function has a recognizable functional inverse.

Answer 1

[Sam Chak]

Hi @Muhammad Adillah

The main issue is caused by the product term because it lacks a functional inverse:

.

Perhaps you will gain a clearer understanding when you plot the product and its numerical functional inverse. Complex values are omitted. While the inverse exists (see red curve), it cannot be expressed in terms of standard mathematical functions.

% assume x = 3
a     = -5:0.001:5;
f     = atan(sqrt((a - 1)*(3^2 - 1)./(a + 1))).*sqrt((a + 1)./(a - 1));
idx   = find(f == real(f));
areal = a(idx);
freal = f(f == real(f));

plot(areal, freal, '.', 'markersize', 1), hold on, grid on
plot(freal, areal, '.', 'markersize', 1)
plot(a, a, '--'), hold off
axis equal
axis([-5 5 -5 5]), xlabel a
legend('f(a)', 'f^{-1}(a)', 'location', 'NW')

[Muhammad Adillah]

I see, so the problem were at the given function and not the code then. Thank you very much for the insight