What does it mean to restrict the domain

The domain is a main component of a function
you can think of a function as a triplet
(F, D, R)
F is the function's formula
D is the domain
R is the range

if one of the component changes the function changes and its properties can change dramaticaly

Let's an example
f(x) : R ---> R
f(x) = x^2

with this domain and range f(x) is not invertible, i.e., some elements y of the range has no x in the domain such that f(x) = y
for instance - 1 is in the range but there is no x such that x^2 = - 1

We can restrict the range and take
f : R ---> R+ (R+ is all nonnegative real numbers)

Now all y in the range have an x in the domain such that y = x^2

Actually, there are TWO different x and this isn't a good new

take y = 4
x^2 = 4
is satisfied by x = -2 and x =2
we cannot say that f is invertible

to make f invertible we restrict its domain , too, taking

f : R+ ---> R+

Now f(x) = x^2 is invertible
x^2 = y has only ONE x such that x^2 = y
namely x = sqrt(y)
which, after swapping variables, leads to
y = sqrt(x)

Other examples are trigonometric functions
sin x range is [-1, 1] as x is in the domain R
but it is not invertible

if we consider
sin x : [-pi, pi] ---> [-1, 1]

sin has an inverse that is called arcsin x

Hope I've answered...

consider y=f(x)
the real values of x which don't give y values in real number system are restrict the domain.
Eg:
y=1/x
when x=0. y goes infinity it is not a real number then 0 is restricted the domain of above function.
y=lnx
values x<1 restrict the domain

Remove some part of the domain. For examole sinx is not one-one function on R but restricting to [-pi/2, pi/2] it is one-one etc.