If f(x)=√x , g(x)=2x-3 , then domain of fog(x) is:

The correct answer is Option (1) → \(\left[\displaystyle\frac{3}{2}, \infty\right)\)

The composite function is

    \(fog(x) = f(g(x)) = \sqrt{2x-3}\)

g is defined for every real x, and the square root needs a non-negative input:

    \(2x-3 \ge 0\)

    \( \displaystyle x \ge \frac{3}{2} \)

So the domain is \(\left[\frac{3}{2}, \infty\right)\).

Why the other options are incorrect

• Option (2) \(\left[\displaystyle\frac{1}{2}, \infty\right)\) — Incorrect. It contains x = 1, where 2x − 3 = −1 and the square root is not real.

• Option (3) \(\left(\displaystyle\frac{3}{2}, \infty\right)\) — Incorrect. It leaves out x = 3/2, where \(fog\left(\frac{3}{2}\right) = \sqrt{0} = 0\) is defined.

• Option (4) \(\left[\displaystyle\frac{5}{2}, \infty\right)\) — Incorrect. It leaves out points such as x = 2, where \(fog(2) = \sqrt{1} = 1\) is defined.