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funtcion

Source: 2023 AMC 12B #22

November 15, 2023
function2023 AMC2023 AMC 12BAMCAMC 12

Problem Statement

A real-valued function ff has the property that for all real numbers aa and b,b, f(a+b)+f(ab)=2f(a)f(b).f(a + b) + f(a - b) = 2f(a) f(b). Which one of the following cannot be the value of f(1)?f(1)? <spanclass=latexbold>(A)</span>0<spanclass=latexbold>(B)</span>1<spanclass=latexbold>(C)</span>1<spanclass=latexbold>(D)</span>2<spanclass=latexbold>(E)</span>2 <span class='latex-bold'>(A) </span> 0 \qquad <span class='latex-bold'>(B) </span> 1 \qquad <span class='latex-bold'>(C) </span> -1 \qquad <span class='latex-bold'>(D) </span> 2 \qquad <span class='latex-bold'>(E) </span> -2