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The arctangent function \(\arctan x\) is the inverse of the tangent function but only over a specific interval. This designer interval allows for a single value output for each input. The behavior of \(\arctan x\) is unique because it transforms infinite \(x\) values to finite \([-\pi/2, \pi/2]\) range values. This function is continuous, which means it has no breaks or gaps, and it is also odd, ensuring symmetry about the origin. After modification to \(\arctan 2x\), the graph becomes steeper. This means the rate at which it climbs changes, due to multiplying the \(x\) value by 2 before applying the arctangent operation. The steepness shows two things:

• The output values change more significantly over small \(x\) changes compared to \(\arctan x\).
• Transitions quickly occur between approaching \(-\pi/2\) and \(\pi/2\).

Understanding behavior helps sketch and predict graph tendencies, which are key for graph interpretation.

For \(\arctan x\), the domain includes all real numbers \(x\), expressed as \(-\infty, \infty\). This means you can input any real \(x\), and the function will output a valid \(y\) value. When we look at \(\arctan 2x\), the domain remains unchanged, hence all real numbers still apply. This is because multiplicating \(x\) by 2 does not restrict any potential input value.
The range of \(\arctan x\) is limited to \([-\pi/2, \pi/2]\), producing only values within this interval. The function stretches from the smallest value, just above \(-\infty\), to the largest, just below \(\infty\), capping at these specific angles given by \(\arctan\). Hence, \(\arctan 2x\) shares this range. Knowing domain and range is crucial for understanding the extent and limitations of function output given any input.

Start sketching \(f(x) = \arctan 2x\) by identifying key features and values, like intercepts and asymptotic behavior. Here's a simple roadmap:

• Intercepts: The function passes through the origin \(0,0\) because \(f(0) = \arctan 0 = 0\).
• Asymptotic Behavior: As \(x\) approaches positive infinity, \(f(x)\) nears \(\pi/2\), and as \(x\) approaches negative infinity, \(f(x)\) nears \(-\pi/2\).

The graph contextually summarizes by hugging these horizontal lines (asymptotes) without crossing them. The middle segment of the graph is vertical due to the factor of 2, steepening it. Remember, \(\arctan 2x\)\ peaks at \(\pi/2\) and bottoms out at \(-\pi/2\).
To summarize, denote the smooth increasing rise from left to right, capturing all behavior within given constraints and ensuring you mark it steeper than the standard \(\arctan x\)\. This visualization aids analytical and computational problem-solving, cementing comprehension.