A fn is symmetric about the y axis if f(-x) = f(x)

in other words if you replace x with -x you get the same formula

For eg y = x^2 = (-x)^2

for y= x^(1/5) -> (-x)^1/5 = -x^(1/5) so it is not symmetric about yaxis

if f(-x) = - f(x) then the fn is symmetric wrt the origin

again this means if you replace x with -x you get -y

for y= x^(1/5) -> (-x)^1/5 = -x^(1/5) = - y

if neither happens you get neither I'll leave the second for you to do

For the domain and range :

y = |x| -2 the domain of |x| is all x and the range is [0,inf)

so for |x| - 2 it is simply a vertical shift of 2 units down so the range

is [-2,inf)

For ln(x) the domain is (0,inf) so ln(x-3) is a horizontal shift to the right

3 so the domain is (3,inf) or if you will the domain is x-3 >0 or x>3

For ln(x) the range is (-inf,inf) so what does adding +1 to the range do?