A polynomial f with real coefficients satisfies f(x) f(y) = f(x) + f(y) + f(xy) - 2 for all real x and y, and f(2) = 5. Which of the following statements are correct?

MathematicsAdvancedHigh School (11–12)Functions and Calculus › Functional Equations JEE Advanced

A polynomial $f$ with real coefficients satisfies $f(x)\,f(y) = f(x) + f(y) + f(xy) - 2$ for all real $x$ and $y$ , and $f(2) = 5$ . Which of the following statements are correct?

One or more options are correct. Tick all the correct ones: a wrong option costs marks, and a part-right answer earns part of the marks.

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← The function f(x) = x² - 4x + 5, taken on the domain [2, ∞), is one-one and maps onto [1, ∞), so it has an inverse f⁻¹ from [1, ∞) to [2, ∞). Which of the following statements are correct?Let A = {1, 2, 3, 4, 5}. How many functions f from A to A satisfy f(f(x)) = f(x) for every x in A? (Such a function does nothing more when it is applied a second time.) →

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