Mathematics

Mathematics

Problem

For the function f(x)=ln⁡(x−1)f(x)=\sqrt{\ln(x-1)}, write the two separate algebraic conditions that xx must satisfy so that f(x)f(x) is defined.
Intuitive explanation

The domain of a composite function is determined by the constraints of each layer. Here, you need the argument of the square root to be non-negative (so the square root is real), and the argument of the logarithm to be positive (so the logarithm exists). These two requirements give you two separate inequalities—write them both out algebraically, and their intersection will be your domain.

Intuitive explanation

The domain of a composite function is determined by the constraints of each layer. Here, you need the argument of the square root to be non-negative (so the square root is real), and the argument of the logarithm to be positive (so the logarithm exists). These two requirements give you two separate inequalities—write them both out algebraically, and their intersection will be your domain.

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