If y = tan^(-1)(x), then the range of y is:

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Question: If y = tan^(-1)(x), then the range of y is:

Options:

Correct Answer: (-π/2, π/2)

Solution:

The range of y = tan^(-1)(x) is (-π/2, π/2).

If y = tan^(-1)(x), then the range of y is:

If y = tan^(-1)(x), then the range of y is:

Step 1: Understand what y = tan^(-1)(x) means. This is the inverse tangent function, which gives the angle whose tangent is x.
Step 2: Recognize that the tangent function can take any real number as input, but its output (the angle) is limited.
Step 3: Know that the tangent function has vertical asymptotes at -π/2 and π/2, meaning it approaches these values but never actually reaches them.
Step 4: Conclude that the output of the inverse tangent function (y) must be between -π/2 and π/2, but not including those values.
Step 5: Therefore, the range of y = tan^(-1)(x) is (-π/2, π/2).

Inverse Trigonometric Functions – Understanding the properties and ranges of inverse trigonometric functions, specifically the arctangent function.
Range of Functions – Identifying the range of a function based on its definition and behavior.