Chapter 1.4, Problem 5QR

Solution

To find: The domain of a function in interval notation.

The domain of the function $f\left(x\right)=\sqrt{\ln \left(x\right)}$ is the interval $\left[1,\infty \right)$ .

Given information: The function is $f\left(x\right)=\sqrt{\ln \left(x\right)}$ .

Method/Formula used:

Suppose $y=f\left(x\right)$ be a function $f:A\to B$ , $x\in A,y\in B$ the set A is called the domain of function f and set B the co-domain. Thus, the domain of a function $f\left(x\right)$ are the values of x for which the function $f\left(x\right)$ is defined.

  

Calculation:

The given function is $f\left(x\right)=\sqrt{\ln \left(x\right)}$ .

Since the logarithm of a negative number is not defined, therefore, the function $\ln \left(x\right)$ is not defined when $x<0.$

Since, the square root of a negative number is an imaginary number, and when $0<x<1$ , the function $\ln \left(x\right)$ is a negative number so $\sqrt{\ln \left(x\right)}$ is not defined. Thus, the function $f\left(x\right)=\sqrt{\ln \left(x\right)}$ is not defined in the interval $\left(-\infty ,1\right)$ , i.e., the function $f\left(x\right)=\sqrt{\ln \left(x\right)}$ is defined in the interval $\left[1,\infty \right)$ that is its domain.

Therefore, the domain D of the function $f\left(x\right)=\sqrt{\ln \left(x\right)}$ is the interval $D=\left[1,\infty \right)$ .