+4 How many solutions does the equation have?Let\[f(x)={−x+3if x≤0,2x−5if x>0.\]
+1729
Ifx≤0, then [f(f(x)) = f(-x + 3) = 2(-x + 3) - 5 = -2x + 1.]If x>0, then [f(f(x)) = f(2x - 5) = -2x - 5 + 3 = -2x - 2.]Hence,our problem is reduced to the two equations −2x+1=4 −2x−2=4.The first equation is solved byx=23,and the second equation is solved byx=−3.Since each solution satisfies the corresponding inequality,there are2solutions.
