(i) Let us find f(5). Since 5 lies between 4 and 6, we have to use f(x) = 3x2 – 10.
Thus, f(5) = 3(52) – 10 = 65.
(ii) To find f(3), note that 3 lies between 2 and 4.
So, we use f(x) = 2x – 1 to calculate f(3).
Thus, f(3) = 2(3) – 1 = 5.
(iii) Let us find f(1).
Now, 1 is in the interval 1 < x < 2
Thus, we have to use f(x) = 1 + x to obtain f(1) = 1 + 1 = 2.
(iv) f(2) – f(4)
Now, 2 is in the interval 2 < x < 4 and so, we use f(x) = 2x – 1.
Thus, f(2) = 2(2) -1 = 3.
Also, 4 is in the interval 4 < x < 6. Thus, we use f(x) = 3x2 – 10
Therefore, f(4) = 3(4 ) – 10 = 3(16) – 10 = 48 – 10 = 38.
Hence, f(2) – f(4) = 3 – 38 = -35.
(v) To calculate 2 f(5) – 3f(1), we shall make use of the values that we have already calculated in (i) and (iii).
Thus, 2f(5) – 3f(1) = 2(65) – 3(2) = 130 – 6 – 124.