Let f(x) = (x - 1)² + 1 for x = 1. Consider: Statement 1: the set {x: f(x) = f⁻¹(x)} = {1, 2}. Statement 2: f is a bijection and f⁻¹(x) = 1 + sqrt(x - 1) for x = 1. Which option is correct?

Question

Let f(x) = (x - 1)² + 1 for x >= 1. Consider: Statement 1: the set {x: f(x) = f⁻¹(x)} = {1, 2}. Statement 2: f is a bijection and f⁻¹(x) = 1 + sqrt(x - 1) for x >= 1. Which option is correct?

Accepted Answer

Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.. f(x) = (x-1)² + 1 is increasing and bijective on [1, infinity) onto [1, infinity), with inverse f⁻¹(x) = 1 + sqrt(x - 1), so Statement 2 is true. Since f is increasing, intersection with its inverse occurs where f(x) = x: (x-1)² + 1 = x gives x² - 3x + 2 = 0, x = 1 or 2, so the set is {1, 2} and Statement 1 is true. Statement 2 (bijection plus the increasing property) correctly explains Statement 1.

Let f(x) = (x - 1)² + 1 for x >= 1. Consider: Statement 1: the set {x: f(x) = f⁻¹(x)} = {1, 2}. Statement 2: f is a bijection and f⁻¹(x) = 1 + sqrt(x - 1) for x >= 1. Which option is correct?

Solution

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