What’s the domain and range of: log(√(2x-1) + 3 ) explain how you got it

Two main facts are needed here:

1. The logarithm , regardless of the base of the logarithm, exists for .

2. The square root exists for .

(in both cases we're assuming real-valued functions only)

By (2) we know that exists if , or .

By (1), we know that exists if , or . But as long as the square root exists, it will always be positive, so this condition will always be met.

Ultimately, then, we only require , so the function has domain .

To determine the range, we need to know that, in their respective domains, and increase monotonically without bound. We also know that at minimum, at which point the square root term vanishes, so the least value the function takes on is . Then its range would be .

horizontal translation of 2 units in the positive x direction

followed by

vertical translation of 3 units in the positive y direction