jmc

Question

For some constants a and b, let f(x) = arraycl ax + b & if x < 2, 8 - 3x & if x 2. array .The function f has the property that f(f(x)) = x for all x. What is a + b?

Accepted Answer

Setting x = 3, we get f(3) = -1. Since -1 < 2, f(-1) = -a + b. Hence, f(f(3)) = f(-1) = -a + b. But f(f(x)) = x for all x, so -a + b = 3. Setting x = 4, we get f(4) = -4. Since -4 < 2, f(-4) = -4a + b. Hence, f(f(4)) = f(-4) = -4a + b. But f(f(x)) = x for all x, so -4a + b = 4. Subtracting the equations -a + b = 3 and -4a + b = 4, we get 3a = -1, so a = -1/3. From -a + b = 3, we get b = a + 3 = 8/3. Hence, a + b = (-1/3) + 8/3 = 73.