Problem 106 - Algebra and Trigonometry

Question

Without drawing a graph, describe the behavior of the graph of \(y=\tan ^{-1} x .\) Mention the function's domain and range in your description.

Step-by-Step Solution

Answer

The graph of \( \tan^{-1}x \) increases for all real numbers, moving from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \) as x ranges from negative to positive infinity. The domain of \( \tan^{-1}x \) is \( (-\infty, \infty ) \) and its range is \( (-\frac{\pi}{2}, \frac{\pi}{2}) \).

Step 1: Determine the domain of \( \tan^{-1}x \)

The domain of \( \tan^{-1}x \) is all real numbers, i.e., \( (-\infty, \infty ) \). The reason is that you can take a tangent of any angle, and this will give you a real number.

Step 2: Determine the range of \( \tan^{-1}x \)

The range of \( \tan^{-1}x \) is \( (-\frac{\pi}{2}, \frac{\pi}{2}) \). The inverse tangent function is defined from negative to positive infinity, but it only takes on values from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \). This is because these are the values in which tangent function takes all real values.

Step 3: Analyze the Behavior of \( \tan^{-1}x \)

The function \( \tan^{-1}x \) is increasing for all real numbers. It starts from \( -\frac{\pi}{2} \) and goes to \( \frac{\pi}{2} \) as x moves from negative to positive infinity. The curve of \( \tan^{-1}x \) is a variant of the letter 'S'.