What is the domain of tangent inverse?
Understand the Problem
The question is asking about the range of values for which the inverse tangent function (arctan) is defined. Specifically, it seeks to understand the input values (domain) for which the output of the function is real numbers.
Answer
The inverse tangent function is defined for all real numbers: $x \in (-\infty, \infty)$.
Answer for screen readers
The inverse tangent function (arctan) is defined for all real numbers: $x \in (-\infty, \infty)$.
Steps to Solve
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Identify the function type The inverse tangent function, denoted as $\text{arctan}(x)$ or $\tan^{-1}(x)$, is defined for all real numbers $x$.
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Determine the domain The domain of the inverse tangent function is all real numbers. This means $x$ can take any value from $-\infty$ to $\infty$, i.e., $$ x \in (-\infty, \infty) $$
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Verify the output The output values of $\text{arctan}(x)$ are restricted to a specific range. The values range from $-\frac{\pi}{2}$ to $\frac{\pi}{2}$, but this doesn't affect the domain.
The inverse tangent function (arctan) is defined for all real numbers: $x \in (-\infty, \infty)$.
More Information
The inverse tangent function is commonly used in various fields, including physics and engineering, to find the angle whose tangent is a given number. Its output is useful in situations when angles need to be calculated based on ratio values.
Tips
One common mistake is to assume that the inverse tangent function is only defined for a limited set of numbers, such as positive numbers. Remember that $\text{arctan}(x)$ is valid for all real numbers.
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