I cannot figure out the domain and range of the equation
y= 4 - x^2
and
x= y^2 - 1

the domain of a function is the set of all x values that produce defined y values
the range of a function is the set of all y values that are produced when all values in the domain are substituted for x.

lets look at the first equation.
y=4-x^2
are there any values that x could be that would produce undefined values? Usually undefined values would be things like 1/0. in this case the answer is no. there are no values of x that will create an undefined result. So the domain of this function is all real numbers or (-infinity,+infinity).
Now, if we make x take on all values from -inf to +inf what values does y take on? The trick for this equation is that x^2 will turn negative numbers into positive numbers. So the largest value that y will take on will be when x=0, 4-0^2=4. any other value of x will result in a subtraction from 4. if x is -inf or +inf we'll have 4-inf=-inf. So y can go from (-inf,4]
the domain is (-inf,+inf) the range is (-inf,4] (if you don't understand the ( and [ let us know and we can explain that too).

why don't you try the second equation and show your work and we can comment on your results.

Thank you!