In order to find the domain and range, we need to find the values of x for which the function is defined.
Since we have square roots, we know that they both need to be greater than or equal to 0. Since the first square root is a subtraction, it is more restrictive so it will control the domain.
sqrt(x-3) >= 0
x-3 >= 0
x >= 3
therefore D_f= x >= 3
Now the range is a bit more tricky to find, but if we look at out function, we see that there's a subtraction happening, and this is key to finding the range of x.
If we look at what is being subtracted, it's sqrt(x-3)-sqrt(x+3). By looking, we know that this subtraction will always be negative because sqrt(x+3) > sqrt(x-3). Taking the smallest valid value of x gives us sqrt0-sqrt6=-sqrt6. Thus, our function will always be greater than or equal to -sqrt6.
Now, for small values of x, the constants will be significant in deciding the value of f, so f will be more negative as there will be a larger difference between the two square roots. However, as x becomes larger and larger, the constants become less significant and the difference between the square roots becomes smaller. Thus f becomes less negative and approaches 0.
However, we know that sqrt(x-3) != sqrt(x+3), so the function will never touch 0. Thus 0 is the limiting value of f.
R_f=-sqrt6 <= f(x) < 0
