What is the domain and range of ## y=sqrt(x^2-1)##?

Domain: ##(-00, -1] uu [1, + oo)## Range: ##[0, + oo)##

The domain of the function will be determined by the fact that the expression that's under the radical must be positive for real numbers.

Since ##x^2## will always be positive regardless of the sign of ##x##, you need to find the values of ##x## that will make ##x^2## smaller than ##1##, since those are the only values that will make the expression negative.

So, you need to have

##x^2 - 1 >=0##

##x^2 >=1##

Take the square root of both sides to get

##|x| >= 1##

This of course means that you have

##x >= 1" "## and ##" "x<=-1##

The domain of the function will thus be ##(-00, -1] uu [1, + oo)##.

The range of the function will be determined by the fact that the square root of a real number must always be positive. The smallest value the function can take will happen for ##x = -1## and for ##x=1##, since those values of ##x## will make the radical term equal to zero.

##sqrt((-1)^2 -1) = 0" "## and ##" "sqrt((1)^2 -1 ) = 0##

The range of the function will thus be ##[0, + oo)##.

graph{sqrt(x^2-1) [-10, 10, -5, 5]}