Work out domain and range instantly with clear inputs, formula shown and shareable results.
The domain is every input the function accepts and the range every output it produces. Restrictions come from three sources: no division by zero, no square root of a negative, and no logarithm of a non-positive number.
Quadratic range
vertex k = c − b²/(4a); range is y ≥ k when a > 0 and y ≤ k when a < 0
Square root domain
ax + b ≥ 0
Domain is all reals; the vertex is at y = 3 − 64/8 = −5, so the range is y ≥ −5.
Because a/(x + b) can be made arbitrarily small but never exactly zero for finite x.