To find the domain of the function $f(x) = 1 - \ln(x)$, we need to consider the properties of the natural logarithm function, $\ln(x)$.

Key property of the natural logarithm:

• The natural logarithm $\ln(x)$ is only defined for positive real numbers. In other words, $\ln(x)$ is defined if and only if $x > 0$.

Thus, the expression $\ln(x)$ in $1 - \ln(x)$ implies that $x$ must be greater than 0.

Domain of $f(x)$:

The domain of $f(x) = 1 - \ln(x)$ is:

$x > 0$

In interval notation, the domain is: $(0, \infty)$

Summary:

The function $1 - \ln(x)$ is defined for all $x$ in the interval $(0, \infty)$.

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Would you like more details on the logarithmic function's properties, or do you have any questions?

Related Questions:

1. What is the range of the function $f(x) = 1 - \ln(x)$?
2. How would the domain change if the function was $1 + \ln(x)$?
3. What happens to the graph of $f(x) = 1 - \ln(x)$ as $x$ approaches infinity?
4. How do you find the inverse of the function $f(x) = 1 - \ln(x)$?
5. What is the derivative of $f(x) = 1 - \ln(x)$?

Tip:

Always remember that the natural logarithm is undefined for non-positive values of $x$. So the domain will always exclude zero and negative numbers.