A function f: R - [-pi/4, pi/2) is defined by f(x) = tan⁻¹(x⁴ - x² - 7/4 + tan⁻¹(alpha)). If f is surjective, which of the following is/are correct?

Question

A function f: R -> [-pi/4, pi/2) is defined by f(x) = tan⁻¹(x⁴ - x² - 7/4 + tan⁻¹(alpha)). If f is surjective, which of the following is/are correct?

Accepted Answer

cos⁻¹((1 - alpha²)/(1 + alpha²)) = 2. For f to be surjective onto [-pi/4, pi/2), the inner expression g(x) = x⁴ - x² - 7/4 + tan⁻¹(alpha) must achieve the minimum value of -1 (since tan(-pi/4) = -1) and be unbounded above. The minimum of x⁴ - x² at x² = 1/2 is -1/4, so min g = -1/4 - 7/4 + tan⁻¹(alpha) = -2 + tan⁻¹(alpha) = -1, giving tan⁻¹(alpha) = 1, i.e., alpha = tan(1). Options A, B, and C are all equivalent identities for alpha = tan(1) and are correct; option D is incorrect.

A function f: R -> [-pi/4, pi/2) is defined by f(x) = tan⁻¹(x⁴ - x² - 7/4 + tan⁻¹(alpha)). If f is surjective, which of the following is/are correct?

Solution

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