Find The Domain Of The Function F ( X ) = 8 Ln ⁡ ( X ) F(x)=8 \sqrt{\ln (x)} F ( X ) = 8 Ln ( X ) ​ .(Give Your Answer As An Interval In The Form (,). Use The Symbol ∞ \infty ∞ For Infinity, U U U For Combining Intervals, And An Appropriate Type Of Parenthesis (,),

**Find the Domain of the Function $f(x)=8 \sqrt{\ln (x)}$**

Understanding the Domain of a Function

The domain of a function is the set of all possible input values (x) for which the function is defined and produces a real number as output. In other words, it is the set of all possible values of x that can be plugged into the function without causing any mathematical errors or undefined values.

The Function $f(x)=8 \sqrt{\ln (x)}$

The given function is $f(x)=8 \sqrt{\ln (x)}$. To find the domain of this function, we need to consider the restrictions imposed by the square root and the natural logarithm functions.

Restrictions Imposed by the Square Root Function

The square root function is defined only for non-negative real numbers. This means that the expression inside the square root, $\ln (x)$, must be greater than or equal to zero.

Restrictions Imposed by the Natural Logarithm Function

The natural logarithm function, $\ln (x)$, is defined only for positive real numbers. This means that the input value, x, must be greater than zero.

Combining the Restrictions

To find the domain of the function $f(x)=8 \sqrt{\ln (x)}$, we need to combine the restrictions imposed by the square root and the natural logarithm functions.

Step 1: Find the values of x that satisfy the restriction imposed by the natural logarithm function

The natural logarithm function is defined only for positive real numbers. Therefore, we need to find the values of x that are greater than zero.

Step 2: Find the values of x that satisfy the restriction imposed by the square root function

The square root function is defined only for non-negative real numbers. Therefore, we need to find the values of x that satisfy the inequality $\ln (x) \geq 0$.

Solving the Inequality

To solve the inequality $\ln (x) \geq 0$, we can use the fact that the natural logarithm function is an increasing function. This means that the inequality is equivalent to $x \geq e^0$, where e is the base of the natural logarithm.

Simplifying the Inequality

The inequality $x \geq e^0$ is equivalent to $x \geq 1$.

Combining the Restrictions

To find the domain of the function $f(x)=8 \sqrt{\ln (x)}$, we need to combine the restrictions imposed by the natural logarithm and the square root functions.

The Domain of the Function

The domain of the function $f(x)=8 \sqrt{\ln (x)}$ is the set of all values of x that satisfy the inequality $x > 0$ and $x \geq 1$.

Simplifying the Domain

The domain of the function $f(x)=8 \sqrt{\ln (x)}$ can be simplified to $x \in (1, \infty)$.

Conclusion

In conclusion, the domain of the function $f(x)=8 \sqrt{\ln (x)}$ is the set of all values of x that are greater than zero and greater than or equal to one. This can be represented as the interval $(1, \infty)$.

Frequently Asked Questions

Q: What is the domain of the function $f(x)=8 \sqrt{\ln (x)}$?

A: The domain of the function $f(x)=8 \sqrt{\ln (x)}$ is the set of all values of x that are greater than zero and greater than or equal to one.

Q: Why is the domain of the function $f(x)=8 \sqrt{\ln (x)}$ restricted to values greater than zero?

A: The domain of the function $f(x)=8 \sqrt{\ln (x)}$ is restricted to values greater than zero because the natural logarithm function is defined only for positive real numbers.

Q: Why is the domain of the function $f(x)=8 \sqrt{\ln (x)}$ restricted to values greater than or equal to one?

A: The domain of the function $f(x)=8 \sqrt{\ln (x)}$ is restricted to values greater than or equal to one because the square root function is defined only for non-negative real numbers.

Q: How can I represent the domain of the function $f(x)=8 \sqrt{\ln (x)}$ in interval notation?

A: The domain of the function $f(x)=8 \sqrt{\ln (x)}$ can be represented in interval notation as $(1, \infty)$.

Q: What is the significance of the domain of a function?

A: The domain of a function is the set of all possible input values (x) for which the function is defined and produces a real number as output.