The point (π/4, 1) lies on the graph of y = tan x. Therefore, the point _______ lies on the graph of y = tan⁻¹ x.

검증된 단계별 안내

Understand the problem: The point \( \left( \frac{\pi}{4}, 1 \right) \) lies on the graph of \( y = \tan x \). This means that when \( x = \frac{\pi}{4} \), \( y = 1 \).

Recall the definition of the inverse tangent function \( y = \tan^{-1} x \): it reverses the roles of \( x \) and \( y \) from the original function \( y = \tan x \).

Since \( y = \tan x \) passes through \( \left( \frac{\pi}{4}, 1 \right) \), the inverse function \( y = \tan^{-1} x \) will pass through the point where the coordinates are swapped, i.e., \( (1, \frac{\pi}{4}) \).

Therefore, the point \( (1, \frac{\pi}{4}) \) lies on the graph of \( y = \tan^{-1} x \).

This step completes the understanding that the inverse function's graph is the reflection of the original function's graph across the line \( y = x \).

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan or tan⁻¹, reverse the effect of their corresponding trigonometric functions. For y = tan⁻¹ x, the output is the angle whose tangent is x, effectively swapping the roles of input and output compared to y = tan x.

Function and Inverse Function Relationship

If a point (a, b) lies on the graph of a function f, then the point (b, a) lies on the graph of its inverse function f⁻¹. This means the coordinates are reversed, reflecting the inverse relationship between the two functions.

Properties of the Tangent Function

The tangent function, tan x, maps angles to real numbers and is periodic with period π. Its inverse, arctan, maps real numbers back to angles within the principal range (-π/2, π/2), ensuring a one-to-one correspondence necessary for defining the inverse.